System, method and computer program product for transforming a polynomial equation from a coordinate frame of one tile to a coordinate frame of another tile at a finer level of a hierachy

ABSTRACT

Z-buffer rendering of three-dimensional scenes is made more efficient through a method for occlusion culling by which occluded geometry is removed prior to rasterization. The method uses hierarchical z-buffering to reduce the quantity of image and depth information that needs to be accessed. A separate culling stage in the graphics pipeline culls occluded geometry and passes visible geometry on to a rendering stage. The culling stage maintains its own z-pyramid in which z-values are stored at low precision (e.g., in 8 bits). The efficiency of hierarchical z-buffering is improved through hierarchical evaluation of line and plane equations.

The present application is a divisional of a parent application filed Sep. 8, 2003 under Ser. No. 10/658,224 which, in turn, is a divisional of a parent application filed Jul. 10, 2002 now U.S. Pat. No. 6,768,487 under Ser. No. 10/193,400 which, in turn, is a divisional of a parent application filed Jul. 22, 1998 now U.S. Pat. No. 6,480,205 under Ser. No. 09/121,317, which are all incorporated herein by reference.

BACKGROUND OF THE INVENTION

The invention, generally, relates to computer graphics and, more particularly, to a new and improved method and apparatus for rendering images of three-dimensional scenes using z-buffering.

Rendering is the process of making a perspective image of a scene from a stored geometric model. The rendered image is a two-dimensional array of pixels, suitable for display.

The model is a description of the objects to be rendered in the scene stored as graphics primitives, most typically as mathematical descriptions of polygons together with other information related to the properties of the polygons. Part of the rendering process is the determination of occlusion, whereby the objects and portions of objects occluded from view by other objects in the scene are eliminated.

As the performance of polygon rendering systems advances, the range of practical applications grows, fueling demand for ever more powerful systems capable of rendering ever more complex scenes. There is a compelling need for low-cost high-performance systems capable of handling scenes with high depth complexity, i.e., densely occluded scenes (for example, a scene in which ten polygons overlap on the screen at each pixel, on average).

There is presently an obstacle to achieving high performance in processing densely occluded scenes. In typical computer graphics systems, the model is stored on a host computer which sends scene polygons to a hardware rasterizer which renders them into the rasterizer=s dedicated image memory. When rendering densely occluded scenes with such systems, the bandwidth of the rasterizer=s image memory is often a performance bottleneck.

Traffic between the rasterizer and its image memory increases in approximate proportion to the depth complexity of the scene. Consequently, frame rate decreases in approximate proportion to depth complexity, resulting in poor performance for densely occluded scenes.

A second potential bottleneck is the bandwidth of the bus connecting the host and the rasterizer, since the description of the scene may be very complex and needs to be sent on this bus to the rasterizer every frame. Although memory and bus bandwidth has been increasing steadily, processor speed has been increasing faster than associated memory and bus speeds.

Consequently, bandwidth limitations can become relatively more acute over time. In the prior art, designers of hardware rasterizers have addressed the bottleneck between the rasterizer and its image memory in two basic ways: increasing image-memory bandwidth through interleaving and reducing bandwidth requirements by using smart memory.

Interleaving is commonly employed in high-performance graphics work stations. For example, the SGI Reality Engine achieves a pixel fill rate of roughly 80 megapixels per second using 80 banks of memory.

An alternative approach to solving the bandwidth problem is called, the smart memory technique. One example of this technique is the Pixel-Planes architecture. The memory system in this architecture takes as input a polygon defined by its edge equations and writes all of the pixels inside the polygon, so the effective bandwidth is very high for large polygons.

Another smart-memory approach is “FBRAM,” a memory-chip architecture with on-chip support for z-buffering and compositing. With such a chip, the read-modify-write cycle needed for z-buffering can be replaced with only writes, and as a result, the effective drawing bandwidth is higher than standard memory.

All of these methods improve performance, but they involve additional expense, and they have other limitations. Considering cost first, these methods are relatively expensive which precludes their use in low-end PC and consumer systems that are very price sensitive.

A typical low-cost three-dimensional rasterization system consists of a single rasterizer chip connected to a dedicated frame-buffer memory system, which in turn consists of a single bank of memory. Such a system cannot be highly interleaved because a full-screen image requires only a few memory chips (one 16 megabyte memory chip can store a 1024 by 1024 by 16 bit image), and including additional memory chips is too expensive.

Providing smart memory, such as FBRAM, is an option, but the chips usually used here are produced in much lower volumes than standard memory chips and are often considerably more expensive. Even when the cost of this option is justified, its performance can be inadequate when processing very densely occluded scenes.

Moreover, neither interleaving nor smart memory addresses the root cause of inefficiency in processing densely occluded scenes, which is that most work is expended processing occluded geometry. Conventional rasterization needs to traverse every pixel on every polygon, even if a polygon is entirely occluded.

Hence, there is a need to incorporate occlusion culling into hardware renderers, by which is meant culling of occluded geometry before rasterization, so that memory traffic during rasterization is devoted to processing only visible and nearly visible polygons. Interleaving, smart memory, and occlusion culling all improve performance in processing densely occluded scenes, and they can be used together or separately.

While occlusion culling is new to hardware for z-buffering, it has been employed by software rendering algorithms. One important class of such techniques consists of hierarchical culling methods that operate in both object space and image space. Hierarchical object-space culling methods include the “hierarchical visibility” algorithm which organizes scene polygons in an octree and traverses octree cubes in near-to-far occlusion order, culling cubes if their front faces are occluded. A similar strategy for object-space culling that works for architectural scenes is to organize a scene as rooms with “portals” (openings such as doors and windows), which permits any room not containing the viewpoint to be culled if its portals are occluded.

Both the hierarchical visibility method and the “rooms and portals” method require determining whether a polygon is visible without actually rendering it, an operation that will be referred to as a visibility query or v-query. For example, whether an octree cube is visible can be established by performing v-query on its front faces.

The efficiency of these object-space culling methods depends on the speed of v-query, so there is a need to provide fast hardware support.

Hierarchical image-space culling methods include hierarchical z-buffering and hierarchical polygon tiling with coverage masks, both of which are loosely based on Warnock's recursive subdivision algorithm.

With hierarchical z-buffering, z-buffer depth samples are maintained in a z-pyramid having N×N decimation from level to level (see N. Greene, M. Kass, and G. Miller, “Hierarchical Z-Buffer Visibility,” Proceedings of SIGGRAPH '93, July 1993). The finest level of the z-pyramid is an ordinary z-buffer. At the other levels of the pyramid, each z-value is the farthest z in the corresponding N×N region at the adjacent finer level. To maintain the z-pyramid, whenever a z-value in the finest level is changed, that value is propagated through the coarser levels of the pyramid.

Since each entry in the pyramid represents the farthest visible z within a square region of the screen, a polygon is occluded within a pyramid cell if its nearest point within the cell is behind the corresponding z-pyramid value. Thus, often a polygon can be shown to be occluded by mapping it to the smallest enclosing z-pyramid cell and making a single depth comparison.

When this test fails to cull a polygon, visibility can be established definitively by subdividing the enclosing image cell into an N×N grid of subcells and by comparing polygon depth to z-pyramid depth within the subcells.

Recursive subdivision continues in subcells where the polygon is potentially visible, ultimately finding the visible image samples on a polygon or proving that the polygon is occluded. Since this culling procedure only traverses image cells where a polygon is potentially visible, it can greatly reduce computation and z-buffer memory traffic, compared to conventional rasterization, which needs to traverse every image sample on a polygon, even if the polygon is entirely occluded.

Hierarchical z-buffering accelerates v-query as well as culling of occluded polygons.

Another algorithm that performs image-space culling with hierarchical depth comparisons is described by Latham in U.S. Pat. No. 5,509,110, “Method for tree-structured hierarchical occlusion in image generators,” April, 1996. Although Latham's algorithm does not employ a full-screen z-pyramid, it does maintain a depth hierarchy within rectangular regions of the screen which is maintained by propagation of depth values.

As an alternative to hierarchical z-buffering with a complete z-pyramid, a graphics accelerator could use a two-level depth hierarchy. Systems used for flight-simulation graphics can maintain a “zfar” value for each region of the screen.

The screen regions are called spans and are typically 2×8 pixels. Having spans enables “skip over” of regions where a primitive is occluded over an entire span.

Another rendering algorithm which performs hierarchical culling in image space is hierarchical polygon tiling with coverage masks. If scene polygons are traversed in near-to-far occlusion order, resolving visibility only requires storing a coverage bit at each raster sample rather than a depth value, and with hierarchical polygon tiling, this coverage information is maintained hierarchically in a coverage pyramid having N×N decimation from level to level.

Tiling is performed by recursive subdivision of image space, and since polygons are processed in near-to-far occlusion order, the basic tiling and visibility operations performed during subdivision can be performed efficiently with N×N coverage masks. This hierarchical tiling method can be modified to perform hierarchical z-buffering by maintaining a z-pyramid rather than a coverage pyramid and performing depth comparisons during the recursive subdivision procedure.

This modified version of hierarchical tiling with coverage masks is believed to be the fastest algorithm available for hierarchical z-buffering of polygons. However, for today's processors, such software implementations of this algorithm are not fast enough to render complex scenes in real time.

A precursor to hierarchical polygon tiling with coverage masks is Meagher's method for rendering octrees, which renders the faces of octree cubes in near-to-far occlusion order using a similar hierarchical procedure.

The ZZ-buffer algorithm is another hierarchical rendering algorithm. Although it does not perform z-buffering, it does maintain an image-space hierarchy of depth values to enable hierarchical occlusion culling during recursive subdivision of image space.

Yet another approach to culling has been suggested, one that renders a z-buffer image in two passes and only needs to shade primitives that are visible. In the first pass, all primitives are z-buffered without shading to determine which primitives are visible, and in the second pass, visible primitives are z-buffered with shading to producing a standard shaded image.

Although this suggested approach reduces the amount of work that must be done on shading, it is not an effective culling algorithm for densely occluded scenes because every pixel inside every primitive must be traversed at least once. In fact, this approach does not fall within an acceptable definition for occlusion culling, since it relies on pixel-by-pixel rasterization to establish visibility.

The object-space and image-space culling methods, described above, can alleviate bandwidth bottlenecks when rendering densely occluded scenes. Suppose that a host computer sends polygon records to a graphics accelerator which renders them with hierarchical z-buffering using its own z-pyramid.

Suppose, further, that the accelerator can perform v-query and report the visibility status of polygons to the host. With hierarchical z-buffering, occluded polygons can be culled with a minimum of computation and memory traffic with the z-pyramid, and since most polygons in densely occluded scenes are occluded, the reduction in memory traffic between the accelerator and its image memory can be substantial.

Hierarchical z-buffering also performs v-query tests on portals and bounding boxes with minimal computation and memory traffic, thereby supporting efficient object-space culling of occluded parts of the scene. While hierarchical z-buffering can improve performance, today's processors are not fast enough to enable software implementations of the traditional algorithm to render complex scenes in real time.

Thus there is a need for an efficient hardware architecture for hierarchical z-buffering.

OBJECT AND BRIEF SUMMARY OF THE INVENTION

It is an object of the present invention to provide a new and improved graphics system for rendering computer images of three-dimensional scenes.

Briefly, the preferred embodiment separates culling of occluded geometry from rendering of visible geometry. According to the invention, a separate culling stage receives geometry after it has been transformed, culls occluded geometry, and passes visible geometry on to a rendering stage. This reduces the amount of geometric and image information that must be processed when rendering densely occluded scenes, thereby reducing memory and bus traffic and improving performance.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a block diagram of the preferred embodiment of the invention.

FIG. 2 is an illustration of a z-pyramid organized in 4×4 tiles.

FIG. 3 is a flowchart of the method for rendering a list of polygons.

FIG. 4 is an illustration showing the relationship of bounding boxes to the view frustum in model space.

FIG. 5 is a flowchart of the method for rendering frames with box culling.

FIG. 6 is a flowchart of the method for sorting bounding boxes into layers.

FIG. 7 is a flowchart of the method for processing a batch of bounding boxes.

FIG. 8 is a flowchart of the method for tiling a list of polygons.

FIG. 9 is a flowchart of the method for geometric processing of a polygon.

FIG. 10 is an illustration of a 4×4 tile showing its coordinate frame.

FIG. 11 is a flowchart of the method for tiling a convex polygon.

FIG. 12 is a flowchart of the method for reading an array of z-values.

FIG. 13 is a flowchart of the method for processing an N×N tile.

FIG. 14 is an illustration a 4×4 tile and a triangle.

FIG. 15 is an illustration of nested coordinate frames.

FIG. 16 is a flowchart of the method for updating array zfar_(x).

FIG. 17 is a flowchart of the method for propagating z-values.

FIG. 18 a is an illustration of a view frustum in model space.

FIG. 18 b is an illustration of the coarsest 4×4 tile in a z-pyramid.

FIG. 19 is a flowchart of a method for determining whether a bounding box is occluded by the “tip” of the z-pyramid.

FIG. 20 is a block diagram of data flow within the culling stage.

FIG. 21 is a side view of a 4×4 tile in the z-pyramid.

FIG. 22 a is an illustration of a 4×4 tile covered by two triangles.

FIG. 22 h is an illustration of the coverage mask of triangle Q in FIG. 22 a.

FIG. 22 c is an illustration of the coverage mask of triangle R in FIG. 22 a.

FIG. 23 is a side view of a 4×4 tile in the z-pyramid and two triangles that cover it.

FIG. 24 is a schematic side view of a 4×4 tile in the z-pyramid.

FIG. 25 is a flowchart of the method for updating a mask-zfar tile record.

FIG. 26 is a side view of a cell in the z-pyramid which is covered by three polygons.

FIG. 27 is an outline of the procedure for rendering frames using frame coherence.

FIG. 28 is a flowchart of the method of determining whether the plane of a polygon is occluded within a cell.

FIG. 29 is a flowchart of a “Create Look-Ahead Frame” procedure.

DETAILED DESCRIPTION OF THE INVENTION

One of the key features in the preferred embodiment is to separate culling of occluded geometry from rendering of visible geometry, so that culling operations are optimized independently. According to this feature, a separate culling stage in the graphics pipeline culls occluded geometry and passes visible geometry on to a rendering stage.

The culling stage maintains its own z-pyramid in which z-values are stored at low precision in order to reduce storage requirements and memory traffic. For example, z-values may be stored as 8-bit values instead of the customary 24-bit or 32-bit values.

Alternatively, occlusion information can be stored in novel data structures which require less storage than a z-pyramid consisting of arrays of z-values.

A second, independent method for reducing storage requirements and memory traffic is to use a low-resolution z-pyramid where each z-value in the finest level is a conservative zfar value for a group of image samples.

The novel algorithm presented herein involving hierarchical z-buffering is more efficient and more suitable for hardware implementation than algorithms that have been used previously. The algorithm performs z-buffer tiling hierarchically on N×N regions of image space using a z-pyramid having N×N decimation from level to level to store the depths of previously rendered polygons.

At each cell encountered during hierarchical tiling of a polygon, conservative culling is performed very efficiently by comparing the z-pyramid value to the depth of the plane of the polygon. This routine hierarchically evaluates the line and plane equations describing a polygon using a novel algorithm that does not require general-purpose multiplication (except for set-up computations).

This evaluation method can also be applied to shading and interpolation computations that require evaluation of polynomial equations at samples within a spatial hierarchy. The framework just described is particularly attractive for hardware implementation because of its simplicity and computational efficiency and the fact that image memory is accessed in N×N tiles during the read-compare-write cycle for depth values.

DEFINITIONS

Culling procedures that may fail to cull occluded geometry but never cull visible geometry are defined as conservative.

Z-buffering determines which scene primitive is visible at each sample point on an image raster.

Each sample point on the image raster is defined as an image sample, and the depth at an image sample is called a depth sample.

A z-buffer maintains one depth sample for each point in the image raster. If individual points in the image raster correspond to individual pixels, it is referred to as point sampling.

An alternative is to maintain multiple depth samples within each pixel to permit antialiasing by oversampling and filtering.

A cell in the z-pyramid is the region of the screen corresponding to a value in the z-pyramid. Preferably, at the finest level of the z-pyramid, cells correspond to depth samples—depths at pixels when point sampling and depths at subpixel samples when oversampling. At coarser levels of the z-pyramid, cells correspond to square regions of the screen, as with image pyramids in general.

N×N decimation from level to level of the z-pyramid is used. N×N blocks of cells that are implicit in the structure of the z-pyramid are identified as tiles or N×N tiles.

A z-pyramid will sometimes be referred to simply as a pyramid. The term bounding box, sometimes shortened to box, is applied to bounding volumes of any shape, including the degenerate case of a single polygon (thus, the term includes polygonal “portals” employed by some culling methods).

Although the tiling algorithm described herein is adapted for z-buffering of polygons, z-buffering can also be applied to other types of geometric primitives, for example, quadric surfaces.

The term primitive applies to all types of geometric primitives including polygons.

As used herein, the term “object” (or “geometric object”) is more general than the term “primitive” (or “geometric primitive”), since it may refer to a primitive, a bounding box, a face of a bounding box, and so forth.

A primitive, bounding box, or other geometric object is occluded if it is known to be occluded at all image samples that it covers, it is visible if it is known to be visible at one or more image samples, and otherwise, it is potentially visible.

For convenience, in some cases, visible and potentially visible objects are collectively referred to as visible.

Apparatus.

FIG. 1 illustrates a preferred embodiment of the present invention in which the numeral 100 identifies a graphics system for rendering geometric models represented by polygons. The graphics system includes a scene manager 110 which sends scene geometry to a geometric processor 120.

The geometric processor 120, in turn, transforms the geometry to perspective space and sends it on to a culling stage 130, which culls occluded geometry and passes visible polygons to a z-buffer rendering stage 140 which generates the output image 150 which is converted to video format in a video output stage 160.

Both the culling stage 130 and the z-buffer renderer 140 have their own dedicated depth buffers, a z-pyramid 170 in the case of the culling stage 130 and a conventional z-buffer 180 in the case of the z-buffer renderer 140. Preferably, the z-buffer 180 and the finest level of the z-pyramid 170 have the same resolution and the same arrangement of image samples.

A “feedback connection” 190 enables the culling stage 130 to report the visibility status of bounding boxes to the scene manager 110 and, also, to send z-pyramid z-values to the scene manager 110.

The culling stage 130 is optimized for very high-performance culling by performing hierarchical z-buffering using a dedicated z-pyramid 170 in which z-values are stored at low precision (for example, 8 bits per z-value) in order to conserve storage and memory bandwidth.

In addition to storing z-values at low precision, the culling stage 130 may also compute z-values at low precision to accelerate computation and simplify computational logic.

Since z-values in the z-pyramid 170 are stored at low precision, each value represents a small range of depths. Therefore, visibility at image samples is not always established definitively by the culling stage 130.

However, computations within the culling stage 130 are structured so that culling is conservative, meaning that some occluded geometry can fail to be culled but visible geometry is never culled. Visibility at image samples is established definitively by the z-buffer renderer 140, since z-values within its z-buffer 180 are stored at full precision (e.g. 32 bits per z-value).

Because of the difference in depth-buffer precision between the z-buffer 180 and the z-pyramid 170, some potentially visible polygons sent from the culling stage 130 on to the z-buffer renderer 140 may not contribute visible samples to the output image 150.

The total amount of storage required by the low-precision z-pyramid in the culling stage is less than the total amount of storage required by the z-buffer in the rendering stage. For example, if each z-value in a z-pyramid having 4×4 decimation is stored in 8 bits and each z-value in a z-buffer having the same resolution is stored in 32 bits, the number of bits in each z-value in the z-buffer is four times the number of bits in each z-value in the z-pyramid, and the total bits of storage in the z-buffer is approximately 3.75 times the total bits of storage in the z-pyramid.

If instead, each z-value in the z-pyramid is stored in 4 bits, the number of bits in each z-value in the z-buffer is eight times the number of bits in each z-value in the z-pyramid, and the total bits of storage in the z-buffer is approximately 7.5 times the total bits of storage in the z-pyramid.

Within the culling stage 130, hierarchical z-buffering is performed using a hierarchical tiling algorithm which includes a hierarchical method for evaluating the linear equations describing polygons according to the invention.

The advantage of this hierarchical evaluation method is that it does not require general-purpose multiplication, enabling implementation with faster and more compact logic. These aspects of the invention will be described in more detail hereinafter.

To facilitate reading and writing in blocks, the z-pyramid is organized preferably in N×N tiles, as illustrated in FIG. 2 for a three-level pyramid 200 organized in 4×4 tiles. Each tile is a 4×4 array of “cells,” which are samples 202 at the finest level of the pyramid and square regions of the screen 206 at the other levels.

4×4 tiles are preferred over other alternatives, such as 2×2 or 8×8 tiles, because with 16 z-values, 4×4 tiles are large enough for efficient memory access and small enough that the utilization of fetched values is reasonably high.

Within the z-pyramid, tiles are “nested:” an N×N tile at the finest level corresponds to a cell inside its “parent tile” at the next-to-finest level, this parent tile corresponds to a cell inside a “grandparent tile” at the adjacent coarser level, and so forth for all “ancestors” of a given tile.

For example, 4×4 tile 220 corresponds to cell 218 inside parent tile 210, and tile 210 corresponds to cell 208 inside grandparent tile 216. In this example, tile 220 “corresponds to” cell 218 in the sense that tile 220 and cell 218 cover the same square region of the screen.

In FIG. 2, the image raster is a 64×64 array of depth samples 202 arranged in a uniform grid, only part of which is shown to conserve space.

When point sampling, these depth samples correspond to a 64×64 array of pixels. Alternatively, when oversampling with a 4×4 array of depth samples within each pixel, this image raster corresponds to a 16×16 array of pixels. Of course, z-pyramids normally have much higher resolution than illustrated in this example.

Herein, as applied to a z-pyramid, the term resolution means the resolution of the z-pyramid's finest level.

The z-value associated with each cell of a z-pyramid is the farthest depth sample in the corresponding region of the screen. For example, in FIG. 2 the z-value associated with cell 208 is the farthest of the 16 corresponding z-values in tile 210 in the adjacent finer level and, also, is the farthest of the 256 depth samples in the corresponding region of the finest level 212 (this region is a 4×4 array of 4×4 tiles).

Thus, the finest level of the z-pyramid 200 is a z-buffer containing the depth of the nearest primitive encountered so far at each image sample, and the other levels contain zfar values, indicating the depths of the farthest depth samples in the z-buffer within the corresponding square regions of the screen.

Since a z-pyramid has a plurality of levels which are each a depth buffer, it can also be described as a hierarchical depth buffer.

Although the z-pyramid of FIG. 2 is organized in N×N tiles, in general, z-pyramid tiles are not necessarily square and need not have the same number of rows and columns. The illustrated structure of nested squares can be modified to accommodate non-square images of arbitrary resolution by storing values for only cells within a rectangular region of each pyramid level. In FIG. 2 of the drawings, image samples are arranged on a regular grid. Alternatively, samples can be “jittered” to reduce aliasing.

The Scene Manager.

The scene manager 110 is implemented in software running on a host processor. It reads the scene model from memory, maintains geometric data structures for the scene model, and initiates the flow of geometry through the graphics system 100. It also initiates commands, such as those that initialize the output image and depth buffers prior to rendering a frame (all values in the z-buffer 180 and z-pyramid 170 are initialized to the depth of the far clipping plane).

The system is structured to operate with or without “box culling” (culling of parts of the scene that are inside occluded bounding boxes). Preferably, densely occluded scenes are rendered with box culling, since this accelerates frame generation.

Rendering a Scene without Box Culling.

In this mode of operation, the scene manager 110 can send all polygons in the scene through the system in a single stream. Each polygon in the stream is transformed to perspective space by the geometric processor 120, tiled into the z-pyramid 170 by the culling stage 130 and, if not culled by the culling stage 130, z-buffered into the output image 150 by the z-buffer renderer 140. This sequence of operations is summarized in procedure Render Polygon List 300, shown in the flowchart of FIG. 3. According to the procedure 300, the geometric processor 120 receives records for polygons from the scene manager 110 and processes them using procedure Transform & Set Up Polygon 900 (step 302), which transforms each polygon to perspective space and performs “set-up” computations.

Transform & Set Up Polygon 900 also creates two records for each polygon, a tiling record containing geometric information that the culling stage 130 needs to perform hierarchical tiling, and a rendering record containing the information needed by the z-buffer renderer 140 to render the polygon. The geometric processor 120 outputs these records to the culling stage 130.

In step 304 of Render Polygon List 300, the culling stage 130 processes these records using procedure Tile Polygon List 800, which tiles each polygon into the z-pyramid 170 and determines whether it is visible. For each visible polygon, the culling stage 130 sends the corresponding rendering record on to the z-buffer renderer 140, which renders the polygon into the output image 150 using conventional z-buffering (step 306). When all polygons have been processed, the output image is complete.

Procedures Transform & Set Up Polygon 900 and Tile Polygon List 800 will be described in more detail later.

Rendering a Scene with Box Culling.

To render a scene with box culling, the scene is organized in bounding boxes having polygonal faces. Before processing the geometry inside a box, the box is tested for occlusion, and if it is occluded, the geometry contained in the box is culled. Box culling can accelerate rendering a great deal.

Processing the boxes in a scene in near-to-far order maximizes culling efficiency and minimizes computation and memory traffic. One way to facilitate near-to-far traversal is to organize polygons into a spatial hierarchy such as an octree. However, building and maintaining a spatial hierarchy complicates the software interface and requires additional storage.

Another way to achieve favorable traversal order is to sort boxes into strict near-to-far order at the beginning of a frame. However, this method requires considerable computation when there are numerous boxes. The preferred embodiment employs a unique ordering system that quickly sorts the boxes into approximate near-to-far order.

The unique ordering system of the invention is illustrated in FIG. 4, which shows the bounding box 400 of all scene geometry within the model-space coordinate frame 402, the view frustum 404 (which is oriented so that four of its faces are perpendicular to the page, for ease of illustration), six bounding boxes labeled A-F, and nine “layers” L0, L1, . . . , L8 defined by planes 406 that are parallel to the far clipping plane 408.

The planes 406 appear as lines in the illustration because they are perpendicular to the page. The planes 406 pass through equally spaced points (e.g. 410, 412) on the line 414 that is perpendicular to the far clipping plane 408 and passes through the corner of model space 416 that is farthest in the “near direction,” where the near direction is the direction of the outward-pointing normal 418 to the “near” face 426 of the view frustum 404. The plane through the nearest corner 416 of model space is called Pnear 424, where the “nearest corner” of a box is the corner which lies farthest in the near direction.

Procedure Render Frames with Box Culling 500, illustrated in FIG. 5 of the drawings, is used to render a sequence of frames with box culling. In step 502, scene polygons are organized into bounding boxes, each containing some manageable number of polygons (e.g., between 50 and 100).

The record for each box includes a polygon list, which may be a list of pointers to polygons rather than polygon records. If a particular polygon does not fit conveniently in a single box, the polygon's pointer can be stored with more than one box. Alternatively, the polygon can be clipped to the bounds of each of the boxes that it intersects.

Next, step 504 begins the processing of a frame by clearing the output image 150, the z-pyramid 170, and the z-buffer 180 (z-values are initialized to the depth of the far clipping plane).

Next, at step 505, viewing parameters for the next frame to be rendered are obtained.

Then, procedure Sort Boxes into Layers 600 organizes the bounding boxes into “layers,” the record for each layer including the boxes whose “nearest corner” lies within that layer. Sort Boxes into Layers 600 also makes a list of boxes that intersect the near face of the view frustum. Boxes on this “near-box list” are known to be visible.

Next, step 506 loops over all boxes on the near-box list and renders the polygon list of each box with Render Polygon List 300. Next, step 508 processes layers in near-to-far order, processing the boxes on each layer's list as a “batch” with Process Batch of Boxes 700, which tests boxes for visibility and renders the polygons in visible boxes.

The advantage of processing boxes in batches rather than one at a time is that visibility tests on boxes take time, and the more boxes that are tested at a time, the less the latency per box. Actually, it is not necessary to process each layer as a single batch, but when organizing boxes into batches, layer lists should be utilized to achieve approximate near-to-far traversal.

When all boxes have been processed, the output image is complete so the image is displayed at step 510 and control returns to step 504 to process the next frame.

Procedure Sort Boxes into Layers 600, illustrated in FIG. 6 of the drawings, maintains a list of boxes for each layer. First, step 602 clears the near-box list and the list for each layer to the null list. While boxes remain to be processed (step 604), step 606 determines the bounds of polygons within the box in the current frame.

Actually, this is only necessary when the box contains “moving” polygons, since the bounds of boxes containing only static polygons can be computed before processing the first frame.

Next, step 608 determines whether the box lies outside the view frustum. One fast way to show that a box lies outside the view frustum is to show that it lies entirely outside a face of the frustum. This can be done by substituting one corner of the box into the face's plane equation.

In FIG. 4, for example, the fact that box F's “nearest corner” 422 lies outside the frustum's “far” face 408 establishes that the box lies outside the frustum. The nearest corners of the boxes are marked with a dot in FIG. 4.

If the box is determined to lie outside the frustum at step 608, control returns to step 604. Otherwise, step 610 determines whether the box intersects the “near” face of the view frustum. If so, the box is added to the near-box list at step 612 and control returns to step 604.

If the box does not intersect the near face of the view frustum, control proceeds to step 614. Step 614 determines the index L of the layer containing the box's nearest corner C using the following formula: L=floor(K*d/dfar), where K is the number of layers, d is the distance from point C to plane Pnear 424, dfar is the distance from plane Pnear 424 to the far clipping plane 408, and floor rounds a number to the nearest smaller integer.

For example, in FIG. 4 zfar is labeled, as is depth d for the nearest corner 420 of box E. In this case, the above formula would compute a value of 5 for L, corresponding to layer L5.

Next, in step 616, the box is added to the list for layer L and control returns to step 604. When step 604 determines that all boxes have been processed, the procedure terminates at step 618.

In FIG. 4, Sort Boxes into Layers 600 places boxes A and B in the near-box list, places box C into the list for layer L4, places boxes D and E into the list for layer L5, and culls box F.

In practice, complex scenes contain numerous boxes and layers typically contain many more boxes than in this example, particularly toward the back of the frustum, which is wider. Also, many more layers should be used than shown in this example to improve the accuracy of depth sorting.

Although the boxes in this example are rectangular solids, a box can be defined by any collection of convex polygons.

In summary, procedure Render Frames with Box Culling 500 is an efficient way to achieve approximately near-to-far traversal of boxes without sorting boxes into strict occlusion order or maintaining a spatial hierarchy.

Processing a Batch of Boxes.

At step 508 of Render Frames with Box Culling 500, the scene manager 110 organizes boxes into batches and calls procedure Process Batch of Boxes 700 (FIG. 7) to process each batch. Within Process Batch of Boxes 700, the scene manager 110 culls boxes which are occluded by the “tip” of the z-pyramid and sends the remaining boxes to the geometric processor 120, which transforms the boxes and sends them to the culling stage 130, which determines the visibility of each box and reports its status to the scene manager 110 on the feedback connection 190. When this visibility information is sent, the “tip” of the z-pyramid is also sent to the scene manager 110 on the feedback connection 190.

Then, for each visible box, the scene manager 110 sends the box's list of polygons out to be rendered, and if boxes are nested, processes the “child” boxes that are inside each visible box using this same procedure. This cycle of operations, which alternates between processing in v-query mode when testing boxes for visibility and processing in rendering mode when rendering scene polygons, continues until the whole scene has been rendered.

Considering now the steps of procedure Process Batch of Boxes 700 (FIG. 7), in step 702, the scene manager 110 tests each box in the batch to see if it is occluded by the tip of the z-pyramid using procedure Is Box Occluded by Tip 1900, which will be discussed later. Occluded boxes are removed from the batch. Next, the scene manager 110 sends records for the front faces of each box in the batch to the geometric processor 120.

Using procedure Transform & Set Up Polygon 900, the geometric processor 120 transforms each face to perspective space and performs the other geometric computations required to create the tiling record for the face, which is then output to the culling stage 130 (step 704). While boxes remain to be processed (step 706), the visibility of each box is established by the culling stage 130, which determines whether its front faces contain at least one visible sample using procedure Tile Polygon List 800 operating in v-query mode (step 708).

If step 708 establishes that the box is visible, the corresponding “v-query status bit” is set to visible in step 710; otherwise, it is set to occluded in step 712. As indicated by step 706, this sequence of steps for processing boxes continues until all boxes in the batch have been processed.

Then, step 714 sends the v-query status bits for the batch of boxes from the culling stage 130 to the scene manager 110 on the feedback connection 190. Next, step 716 copies the tip of the z-pyramid to the scene manager 110 on the feedback connection 190. The “tip” includes the farthest z-value in the pyramid, the coarsest N×N tile in the pyramid, and perhaps some additional levels of the pyramid (but not the entire pyramid, since this would involve too much work).

If the farthest z-value in the z-pyramid is nearer than the depth of the far clipping plane maintained by the scene manager 110, step 716 resets the far clipping plane to this farthest z-value. Copying the tip of the pyramid enables the scene manager 110 to cull occluded boxes at step 702, as will be described later.

Next, the scene manager 110 checks the v-query status of each box in the batch and initiates processing of the geometry inside each visible box (step 718). In step 720, the list of polygons associated with a visible box is rendered with procedure Render Polygon List 300.

According to procedure Render Frames with Box Culling 500, bounding boxes are not nested, but nested bounding boxes can be handled with recursive calls to Process Batch of Boxes 700, as indicated by optional steps 722 and 724. If there are “child” boxes inside the current box (step 722), in step 724 the scene manager 110 organizes these boxes into one or more batches and processes each batch with this same procedure 700.

Preferably, batches are processed in near-to-far order, since this improves culling efficiency. When all child boxes have been processed (or if there are no child boxes), control returns to step 718, and when all visible boxes have been processed the procedure 700 terminates at step 726.

Culling with the Z-Pyramid.

Tile Polygon List 800 (FIG. 8) is the procedure used by the culling stage 130 to tile a list of polygons. The procedure 800 receives as input from the geometric processor 120 the processing mode, either v-query or render, and a list of records for polygons.

When in render mode the geometric processor 120 outputs a tiling record for each polygon (geometric information that the culling stage 130 needs to perform hierarchical tiling) and a rendering record for each polygon (information needed by the z-buffer renderer 140 to render the polygon). When in v-query mode, the geometric processor 120 only outputs a tiling record for each polygon.

Tile Polygon List 800 operates in render mode at step 304 of procedure Render Polygon List 300, and it operates in v-query mode at step 708 of procedure Process Batch of Boxes 700.

While polygons remain to be processed (step 802), Tile Polygon List 800 passes the processing mode, a tiling record and, if in render mode, a rendering record to Tile Convex Polygon 1100, the hierarchical tiling procedure employed by the culling stage 130. When in v-query mode, this procedure 1100 just determines whether the polygon is visible with respect to the z-pyramid 170.

When in render mode, the procedure 1100 updates the z-pyramid 170 when visible samples are encountered, and if the polygon is visible, outputs its rendering record to the z-buffer renderer 140. At step 804, if in v-query mode and the polygon is visible, step 806 reports that the polygon list is visible and the procedure terminates at step 808.

Otherwise, the procedure returns to step 802 to process the next polygon. If the procedure 800 is still active after the last polygon in the list has been processed, if in v-query mode at step 810, step 812 reports that the polygon list is occluded and then the procedure terminates at step 814.

Instead, if in render mode at step 810, the procedure terminates immediately at step 814.

Tiling Records.

Geometric computations are performed on polygons by the geometric processor 120 using procedure Transform & Set Up Polygon 900 (FIG. 9). This procedure 900 is employed in step 302 of procedure Render Polygon List 300 and also in step 704 of Process Batch of Boxes 700.

For each polygon, Transform & Set Up Polygon 900 receives input from the scene manager 110 in the form of a record for the polygon before it has been transformed to perspective space, and for each polygon received, the procedure 900 outputs a tiling record, and when in render mode, it also outputs a rendering record.

First, step 902 transforms the polygon's vertices to perspective space. Next, step 904 determines the smallest N×N tile in the pyramid that encloses the transformed polygon.

For example, in FIG. 2 tile 210 is the smallest enclosing 4×4 tile for triangle 214. (Triangle 214 is also enclosed by 4×4 tile 216, but this tile is considered “larger” than tile 210 because it is larger in screen area—it covers the whole screen, whereas tile 210 covers one-sixteenth of the screen.)

Next, step 906 establishes the corner of the screen where the plane of the polygon is nearest to the viewer (i.e., farthest in the “near” direction). The method for computing this “nearest corner” will be described later, in connection with step 1308 of procedure 1300.

Next, step 908 computes the equation of the plane of the polygon and the equation of each edge of the polygon. The coefficients in these equations are relative to the smallest enclosing N×N tile.

Next, step 910 creates a tiling record for the polygon from the geometric information computed in the preceding steps and outputs this record to the culling stage 130. If in render mode, step 910 also creates a rendering record for the polygon which contains the information needed by the z-buffer renderer 140 to render the polygon, and outputs this record to the culling stage 130. Following step 910, the procedure terminates at step 912.

Geometric information computed for a polygon by Transform & Set Up Polygon 900 is stored in a tiling record 5000 containing the following information.

Tiling Record.

1. level number and index of smallest enclosing tile (“level,” “index”);

2. screen corner where plane of polygon is nearest (“nearest_corner”);

3. number of edges (“n”);

4. coefficients (A₁, B₁, C₁), . . . (A₂, B₂, C₂), . . . , (A_(n), B_(n), C_(n)) of edge equations (polygon has n edges); and

5. coefficients (A_(p), B_(p), C_(p)) of plane equation.

The level number and index specify the tile in the z-pyramid (“index” is an array index). The numerical values of the coefficients of the edge and plane equations depend on the coordinate frame in which they are computed, and FIG. 10 shows the “standard coordinate frame” that is used for an arbitrary 4×4 tile 1000.

The origin of the coordinate frame is located at the tile's lower-left corner 1002, and the x and y axes 1004 are scaled so that the centers 1006 of cells 1008 correspond to odd integer coordinates and cell borders correspond to even integer coordinates. Thus, if an N×N tile is at the finest level of the pyramid and image samples are arranged on a uniform grid, the coordinates of image samples are the odd integers 1, 3, 5, . . . 2N−1. If an N×N tile is not at the finest level, its cells are squares whose borders lie on the even integers 0, 2, 4, . . . , 2N. The fact that cell coordinates are small integer values simplifies evaluation of line and plane equations.

Each tile in the z-pyramid has an associated coordinate frame positioned and scaled relative to that tile as illustrated in FIG. 10. For example, FIG. 2 shows the coordinate frames (e.g. 222, 224) of the eight 4×4 tiles that would be traversed during hierarchical tiling of triangle 214.

The Algorithm for Hierarchical Z-Buffering.

Within Tile Polygon List 800, the procedure that hierarchically z-buffers a convex polygon is Tile Convex Polygon 1100 (FIG. 11). The input to this procedure 1100 is the processing mode, either render or v-query, a tiling record, and if in render mode, a rendering record.

When in render mode, the procedure 1100 tiles the polygon into the z-pyramid 170, updates z-values when visible samples are encountered, and if the polygon is visible, outputs its rendering record to the z-buffer renderer 140.

When in v-query mode, the polygon is a face of a bounding box and the procedure 1100 determines whether that face contains at least one visible image sample. When in v-query mode, the z-pyramid 170 is never written, and processing stops if and when a visible sample is found.

Now, data structures maintained by Tile Convex Polygon 1100 are described. The procedure 1100 maintains a stack of temporary tile records called the “Tile Stack,” which is a standard “last in, first out” stack, meaning that the last record pushed onto the stack is the first record popped off.

The temporary records in the Tile Stack contain the same information as the tiling records previously described, except that it is not necessary to include the polygon's “nearest corner,” since this is the same for all tiles.

For each level in the pyramid, Tile Convex Polygon 1100 maintains information about the z-pyramid tile within that level that was accessed most recently. Some of this information is relative to the tile currently being processed, the “current tile.” The level record 5100 for level J of the pyramid contains:

Level Record [J].

1. index of corresponding z-pyramid tile, call this tile “T” (“index[J]”);

2. N×N array of z-values for tile T (“z-array[J]”);

3. farthest z-value in z-array[J], excluding cell containing “current tile” (“zfar_(x)[J]”);

4. TRUE/FALSE flag: Is z-array[J] different than z-pyramid record? (“dirty_flag[J]”); and

5. TRUE/FALSE flag: Is tile T an ancestor of current tile? (“ancestor_flag[J]”).

As listed above, the level_record[J] contains the index for the corresponding tile “T” in the z-pyramid (“index[J]”), the N×N array of z-values corresponding to tile T (“z-array[J]”), the farthest z-value in z-array[J], excluding the depth of the cell containing the current tile (“zfar_(x)[J],” where subscript “x” alludes to this exclusion rule), a flag indicating whether the values in z-array[J] differ from the corresponding values in the z-pyramid (“dirty_flag[J]”), and a flag indicating whether tile T is an “ancestor” of the current tile (“ancestor_flag[J]” is TRUE if the current tile lies inside tile T).

For example, assume that indexes 0, 1, . . . , F refer to the coarsest, next-to-coarsest, . . . , finest levels of the pyramid, respectively. In FIG. 2 of the drawings, while processing tile 220, level_record[0] would correspond to the root tile 216, level_record[1] would correspond to tile 210 (since this would be the most recently accessed tile at level 1), and level_record[2] would correspond to tile 220.

As for ancestor flags, ancestor_flag[0] would be TRUE, since tile 216 is the “grandparent” of tile 220 (in fact, ancestor_flag[0] is always TRUE), ancestor_flag[1] is TRUE since tile 210 is the “parent” of tile 220, and ancestor_flag[2] is FALSE, because a tile is not considered to be an ancestor of itself.

According to the algorithm, which will be described later, while processing tile 220, zfar_(x) values are computed for each pyramid level in order to facilitate propagation of z-values when visible samples are found. After processing tile 220, zfar_(x)[0] would be the farthest z-value in tile 216 excluding cell 208 (the cell that contains tile 220), zfar_(x)[1] would be the farthest z-value in tile 210 excluding cell 218 (the cell that contains tile 220), and zfar_(x)[2] would be the farthest of all the z-values in tile 220. Given these zfar_(x) values, at each level of the pyramid, propagation of z-values only requires comparing one or two z-values, as will be described later.

The Tiling Algorithm.

Tile Convex Polygon 1100 starts with step 1102. If in v-query mode, step 1102 initializes the visibility status of the polygon to occluded.

Next, step 1104 initializes the Tile Stack to the tiling record that was input. Ancestor_flags need to be computed when the tile stack is initialized at step 1104. While the Tile Stack is not empty (step 1106), step 1108 gets the record for the next tile to process (the “current tile”) by popping it from the stack (initially, this is the tiling record that was input, which corresponds to the smallest enclosing tile).

The level in the pyramid of the current tile is called “L.” Step 1110 checks to see if the z-values for the current tile are already in z-array[L] (this can be established by comparing the current tile's index to index[L]). If not, procedure Read Z-Array 1200 reads the z-values for the current tile from the z-pyramid 170 and puts them in z-array[L].

Next, Process N×N Tile 1300 processes each of the cells within the current tile, and if L is not the finest level, for each cell where the polygon is potentially visible, appends a new record to the Tile Stack, as will be described later.

At step 1112, if in v-query mode, control proceeds to step 1114, where if the polygon's status is visible (this is determined in Process N×N Tile 1300), the procedure terminates at step 1116, and otherwise, control returns to step 1106.

If in render mode at step 1112, if L is the finest level of the pyramid and the changed flag is TRUE at step 1118 (this flag is set in Process N×N Tile 1300), step 1120 writes z-array[L] to the z-pyramid 170, Propagate Z-Values 1700 “propagates” z-values through the pyramid (if necessary), and control returns to step 1106.

If L is not the finest level of the pyramid or the changed flag is FALSE at step 1118, control returns directly to step 1106. If the Tile Stack is empty at step 1106, hierarchical tiling of the polygon is complete and the procedure terminates at step 1122. If step 1122 is executed when in v-query mode, the polygon is occluded, but since the polygon's visibility status was initialized to occluded at step 1102, it is not necessary to set the status here.

When in render mode, prior to returning at step 1122 the procedure 1100 can output additional information about a visible polygon to the z-buffer renderer 140. For example, if a polygon is being rendered with texture mapping and texture coordinates are computed during tiling, the bounding box of texture coordinates for the polygon could be output to inform the z-buffer renderer 140 which regions of a texture map will need to be accessed.

Summarizing the role of the Tile Stack in Tile Convex Polygon 1100 when operating in render mode, the tile stack is initialized to a tiling record corresponding to the smallest tile in the z-pyramid that encloses the transformed polygon.

Next, a loop begins with the step of testing whether the Tile Stack is empty, and if so, halting processing of the polygon. Otherwise, a tiling record is popped from the Tile Stack, this tile becoming the “current tile.”

If the current tile is not at the finest level of the pyramid, Process N×N Tile 1300 determines the cells within the current tile where the polygon is potentially visible, creates tiling records corresponding to the potentially visible cells and pushes them onto the Tile Stack, and then control returns to the beginning of the loop. If the current tile is at the finest level of the pyramid, Process N×N Tile 1300 determines any visible samples on the polygon, and if visible samples are found, the z-pyramid is updated. Then, control returns to the beginning of the loop.

The basic loop is the same when in v-query mode except that when a visible sample is encountered, the procedure reports that the polygon is visible and then terminates, or if an empty Tile Stack is encountered, the procedure reports that the polygon is occluded and then terminates.

Procedure Tile Convex Polygon 1100 performs hierarchical polygon tiling and hierarchical v-query of polygons by recursive subdivision. The Tile Stack is the key to implementing recursive subdivision with a simple, efficient algorithm that is well suited for implementation in hardware.

The procedure finishes processing one N×N tile before beginning another one, and reads and writes z-values in N×N blocks. These are not features of prior-art software implementations of hierarchical tiling, which use depth-first traversal of the pyramid, processing all “children” of one cell in a tile before processing other cells in the tile.

Thus, with prior-art software methods, the “traversal tree” describing the order in which z-pyramid tiles are traversed is topologically different than with the tiling algorithm presented herein, which is better suited to implementation in hardware.

The following describes the three procedures called by Tile Convex Polygon 1100: Read Z-Array 1200, Process N×N Tile 1300, and Propagate Z-Values 1700.

Procedure Read Z-Array 1200 (FIG. 12) reads the N×N array of z-values corresponding to a tile specified by its level number (“L”) and index (“I”) from the z-pyramid 170 into z-array[L]. At step 1202, if dirty_flag[L] is TRUE (meaning that the values in z-array[L] have been modified), step 1204 writes z-array[L] to the z-pyramid 170, writes I to index[L], sets dirty_flag[L] to FALSE, and sets ancestor_flag[L] to TRUE.

Next, whether or not step 1204 was executed, step 1206 reads z-values for the specified tile from the z-pyramid 170 into z-array[L], and the procedure terminates at step 1208.

Processing of Tiles.

Process N×N Tile 1300 (FIG. 13) loops over each of the N×N cells within a tile, processing them in sequence, for example by looping over the rows and columns of cells within the tile. The tile's level number in the pyramid is called “L” and the cell currently being processed will be called the “current cell.”

If L is the finest level and in render mode, step 1302 sets a flag called changed to FALSE, sets a variable called zfar_finest to the depth of the near clipping plane, and sets all values in array zfar_(x) to the depth of the near clipping plane. While cells remain to be processed (step 1304), if L is the finest level and in render mode, step 1306 updates array zfar_(x) using procedure Update zfar_(x) 1600.

Occlusion Test.

Next, step 1308 determines whether the plane of the polygon is occluded within the current cell. The polygon's plane equation, which is stored in the tiling record, has the form: z=Ax+By+C.

If the current cell corresponds to an image sample, the depth of the polygon is computed at this sample by substituting the sample's x and y coordinates into the polygon's plane equation.

If the polygon's depth at this point is greater than the corresponding z-value stored in z-array[L] (which is maintained in Tile Convex Polygon 1100), this sample on the polygon is occluded, and control proceeds to step 1312. At step 1312, if at the finest level of the pyramid and in render mode, if the z-value in z-array[L] which corresponds to the current cell is farther than variable zfar_finest, variable zfar_finest is overwritten with that z-value. Following step 1312, control returns to step 1304.

At step 1308, if the current cell corresponds to a square region of the screen (rather than an image sample), the nearest point on the plane of the polygon within that square is determined. This is done by evaluating the plane equation at the corner of the cell where the plane is nearest to the viewer.

This “nearest corner” can be determined easily from the plane's normal vector using the following method, which is illustrated in FIG. 14.

Suppose that triangle 1400 is being processed within cell 1402 of tile 1404, and vector 1406 is a backward-pointing normal vector (nx,ny,nz). Then the corner of the cell 1402 corresponding to the “quadrant” of vector (nx,ny) indicates the corner where the plane of the polygon is nearest to the viewer.

In this instance, the “nearest corner” is 1408, since nx and ny are both negative. (In general, the +x,+y quadrant is upper right, the +x,−y quadrant is lower right, the −x,−y quadrant is lower left, and the −x,+y quadrant is upper left.)

To help in visualizing this, the normal vector 1406 attaches to the center of the back of the triangle 1400, points into the page, and the dashed portion is occluded by the triangle 1400. Step 906 of Transform & Set Up Polygon 900 uses this method to compute the polygon's nearest corner, which is the same at all tiles.

In the case that the normal vector is forward-pointing instead of backward-pointing, a cell's nearest corner corresponds to the quadrant of vector (−nx, −ny) instead of vector (nx, ny).

The next step is to compute the depth of the plane of the polygon at the nearest corner of the current cell, called the plane's znear value within the cell, by substituting the corner's x and y coordinates into the polygon's plane equation, which has the form z=Ax+By +C, where x and y are even integers. Actually, this equation is evaluated hierarchically, as will be explained later.

Next, the plane's znear value is compared to the zfar value stored in z-array[L] that corresponds to the current cell, and if the znear value is farther than the zfar value, the plane of the polygon is occluded within the current cell and control proceeds to step 1312. Otherwise, control proceeds to step 1310.

The depth comparison described above is the only occlusion test performed on a polygon with respect to a given cell. This single occlusion test is not definitive when the nearest corner of the cell lies outside the polygon.

In this case, rather than perform further computations to establish visibility definitively, the occlusion testing of the polygon with respect to the cell is halted and visibility is resolved by subdivision. This culling method is preferred because of its speed and simplicity.

The steps of the above method for testing a polygon for occlusion within a cell covering a square region of the screen are summarized in the flowchart of FIG. 28, which describes the steps performed at step 1308 when the current cell corresponds to a square region of the screen (rather than an image sample).

First, step 2802 determines the corner of the cell where the plane of the polygon is nearest using the quadrant of vector (nx, ny), where (nx, ny, nz) is a backward-pointing normal to the polygon (or if the normal is forward-pointing, the quadrant of vector (−nx, −ny) is used instead).

Next, step 2804 computes the depth of the plane at that “nearest corner,” i.e., the plane's znear value. At step 2806, if the plane's znear value is farther than the z-value for the cell stored in the z-pyramid, step 2808 reports that the plane (and hence the polygon) is occluded and the procedure terminates at step 2812.

Otherwise, step 2810 reports that the plane (and hence the polygon) is potentially visible and no further occlusion testing is performed for the polygon with respect to the cell. Following step 2810, the procedure terminates at step 2812.

Examples of occlusion tests performed by procedure Is Plane Occluded within Cell 2800 are illustrated in FIG. 26, which shows a side view of a cell in a z-pyramid, which in three dimensions is a rectangular solid 2600 having a square cross-section. Given the indicated direction of view 2602, the right-hand end 2604 of the solid 2600 is the near clipping plane and the left-hand end 2606 of the solid 2600 is the far clipping plane.

The bold vertical line indicates the current z-value 2608 stored in the z-pyramid cell. The three inclined lines, 2610, 2620, and 2630, indicate the positions of three polygons, each covering the cell and each oriented perpendicular to the page to simplify illustration. For each polygon, the znear and zfar values of its plane within the cell are shown by dashed lines.

Procedure Is Plane Occluded within Cell 2800 would show that polygon 2610 is occluded at the illustrated cell because the znear value 2612 of the polygon's plane is farther than the cell's z-pyramid value 2608. Procedure Is Plane Occluded within Cell 2800 would show that polygon 2620 is potentially visible at the illustrated cell because the znear value 2622 of the polygon's plane is nearer than the cell's z-pyramid value 2608.

It is preferable that z-values within the z-pyramid 170 are stored at low-precision (e.g., in 8 bits), and this complicates depth comparisons slightly. A low-precision z-value can be thought of as representing a small range of z-values in the interval [near far].

If the plane's znear value computed at step 1308 is farther than far, the plane is occluded within the cell, and if znear is nearer than near, the plane is visible within the cell. But if znear is between near and far it cannot be determined whether the plane is visible within the cell.

In this last case, it is assumed that the polygon is visible so that culling will be conservative, never culling a polygon containing a visible image sample. This same analysis is applied in the other conservative culling procedures discussed herein when depth comparisons involving low-precision z-values are performed.

Overlap Tests.

At step 1310 of procedure 1300, the objective is to determine whether the current cell and the polygon overlap on the screen.

There can be no overlap where the current cell lies entirely outside an edge of the polygon. For each of the polygon's edges, it is determined whether the current cell lies outside that edge by substituting the appropriate point into its edge equation, which has the form: Ax+By+C=0.

If the current cell corresponds to an image sample, the “appropriate point” is that image sample.

In FIG. 10, assume that tile 1000 is at the finest level of the pyramid and the half-plane 1012 lying outside edge 1010 is defined by the inequality Ax+By+C<0. Coefficients A, B, and C in this inequality (which were computed at step 908 of procedure 900) are computed relative to the tile's coordinate frame 1004, and image samples within the tile have odd integer coordinates.

To determine whether an image sample lies outside an edge, its x and y coordinates are substituted into the edge's equation and the sign of the result is checked. Step 1310 performs this test on each edge of the polygon (or until it is determined that the sample lies outside at least one edge). If the sample is outside any edge, control proceeds to step 1312. Otherwise, control proceeds to step 1314.

At step 1310, if the current cell corresponds to a square region of the screen (rather than an image sample), it must be determined whether that square lies entirely outside an edge of the polygon. For each edge, this can be done by substituting the coordinates of a single corner point of the current cell into the edge equation, using the corner that is farthest in the “inside direction” with respect to the edge.

In FIG. 10, the inside direction for edge 1010 is indicated by arrow 1018, the corner of cell 1022 that is farthest in the inside direction is corner 1020, and substituting the corner's x and y coordinates into the equation for edge 1010 shows that corner 1020 and cell 1022 lie outside of edge 1010. The corner points of cells have even integer coordinates, (2,2) in the case of point 1020.

Step 1310 determines whether the current cell lies outside any edge of a polygon by using this method to compare the cell to each edge.

This method is not a definitive cell-polygon intersection test, but it is simple and conservative, never culling a cell containing a visible image sample. If the current cell is outside any edge, control proceeds to step 1312. Otherwise, control proceeds to step 1314.

Step 1308 and each of the “outside-edge” tests of step 1310 can all be done in parallel.

At step 1314, if L is the finest level, a visible image sample has been found, and control proceeds to step 1316.

If in v-query mode at step 1316, step 1318 sets the polygon's visibility status to visible and the procedure terminates at step 1320. If not in v-query mode, if the polygon's rendering record has not yet been output to the z-buffer renderer 140, this is done at step 1322.

In the preferred embodiment of the invention, the resolution of the finest level of the z-pyramid 170 is the same as the resolution of the image raster. However, it is also possible to use a low-resolution z-pyramid. This option and associated steps 1334 and 1336 will be described in more detail later.

Assuming a full-resolution z-pyramid 170, following step 1322, step 1326 sets the changed flag to TRUE. Next, step 1328 updates zfar_finest, a variable that keeps track of the farthest z-value encountered thus far within the current tile. Accordingly, if the z-value computed for the current cell at step 1308 is farther than zfar_finest, zfar_finest is overwritten with that z-value.

Next, step 1330 writes the z-value computed for the polygon at step 1308 to the appropriate entry in z-array[F] (where F is the index of the finest pyramid level).

It is possible to update the z-pyramid 170 directly at this step, but to improve efficiency, preferably, the z-pyramid is read and written in records for N×N tiles.

According to the preferred embodiment of the present invention (FIG. 1), shading is not performed in the stage of the graphics system that is presently being described, but it is possible to do so. For example, it is possible to combine the culling stage 130 and its z-pyramid 170 with the z-buffer renderer 140 and its z-buffer 180 into a single stage: a hierarchical z-buffer renderer with a z-pyramid.

With this architecture, step 1332 would compute the color of the image sample and then overwrite the output image. Also, step 1322 would be omitted (as would step 1340), since there would no longer be a separate rendering stage. Step 1332 is shown in a dashed box to indicate that it is an option and not the preferred method.

Whether or not pixels are shaded in this procedure 1300, control returns to step 1304.

At step 1314, if L is not the finest level, control proceeds to step 1338, which is an optional step (as indicated by its depiction in dashed lines). If in render mode, step 1338 computes the maximum amount that continued tiling within the current cell can advance z-values in the pyramid, which is the difference between the znear value of the polygon's plane computed at step 1308 and the z-value stored for the current cell in z-array[L].

If the maximum “z advance” is less than some specified positive threshold value, call it zdelta, the current cell is not subdivided and the polygon is assumed to be visible. In this case, control proceeds to step 1340, which outputs the polygon's rendering record to the z-buffer renderer 140, if this has not already been done, after which control returns to step 1304.

In FIG. 26 the bold dashed line 2640 shows the z-pyramid value 2608 for a cell offset in the near direction by zdelta. Since the znear value 2622 for polygon 2620 is farther than this offset z-pyramid value 2640, tiling of polygon 2620 would stop within the illustrated cell, since the maximum amount that continued tiling could advance the z-pyramid value for the cell is less than zdelta. On the other hand, tiling of polygon 2630 would continue, since its znear value 2632 is nearer than the offset z-pyramid value 2640.

Although step 1338 can decrease the culling efficiency of the z-pyramid, it also reduces the amount of tiling the culling stage 130 needs to do, and in some cases, this is a good trade-off, improving the overall performance of the system.

If step 1338 is not employed or if its conditions are not satisfied, control proceeds to step 1342. Steps 1342 and 1344 create the tiling record for a new N×N tile corresponding to the current cell, this record including new coefficients for the polygon's edge and plane equations.

Step 1342 “transforms” the current tile's edge and plane equations so that their coefficients are relative to the coordinate frame of the new tile, using a method that will be described later. If tiling records also include the coefficients of shading equations, these equations are also transformed.

Step 1344 computes the level number and index of the new tile, creates a tiling record for the tile, and pushes this record onto the Tile Stack. Following step 1344, control returns to step 1304.

When all cells within the tile have been processed at step 1304, the procedure terminates at step 1346.

Although procedure Process N×N Tile 1300 processes cells one by one, it is also possible to process cells in parallel, for example, by processing one row of cells at a time.

Hierarchical Evaluation of Line and Plane Equations.

Before describing the hierarchical evaluation method employed by the invention, the underlying problem will be described. When z-buffering a polygon, it is necessary to evaluate the linear equations defining the polygon's edges and plane.

Edge equations have the form Ax+By+C=0 and plane equations are expressed in the form z=Ax+By+C. When performing hierarchical z-buffering, these equations must be evaluated at points on tiles in the image hierarchy.

Each of these equations includes two additions and two multiplications so direct evaluation is relatively slow, and if evaluation is performed with dedicated hardware, the circuitry required to perform the multiplications is relatively complex.

Efficient evaluation of these equations is the cornerstone of various prior-art algorithms for z-buffering polygons. However, prior-art methods are not particularly efficient when a polygon covers only a small number of samples, as is the case when tiling is performed on tiles of an image hierarchy, and they do not take advantage of coherence that is available in an image hierarchy.

Thus, there is a need for a more efficient method for evaluating the linear equations defining a polygon within tiles of an image hierarchy.

The novel method employed by the invention achieves efficiency by evaluating line and plane equations hierarchically, as will be described now.

Within Process N×N Tile 1300, at every cell it is necessary to evaluate a plane equation of the form z=Ax+By+C at step 1308 and edge equations of the form Ax+By+C=0 at step 1310. Coefficients A, B, and C are computed relative to the standard coordinate frame of FIG. 10, and the advantage of this approach is that the values of x and y in the equations are small integers, which permits the equations to be evaluated with shifts and adds, rather than performing general-purpose multiplication.

For example, while looping over cells within a tile, equation z=Ax+By+C can be computed incrementally as follows: at (0,0) z=C, at (2,0) z=C+2A, at (4,0) z=C+4A, and so forth.

Even when incremental methods are not used, the equations can be evaluated efficiently with shifts and adds.

For example, if x is 5, the term Ax can be computed by adding A to 4A, where 4A is obtained by shifting.

At step 1342 of Process N×N Tile 1300, new coefficients of edge and plane equations are computed when cells are subdivided. The objective is to transform a linear equation of x and y from the coordinate frame of an N×N tile to the coordinate frame of cell (xt,yt) within it.

More particularly, in FIG. 2 consider cell 218 within 4×4 tile 210, which corresponds to 4×4 tile 220 at the adjacent finer level of the pyramid. The relationship in screen space between the (x,y) coordinate frame 222 of cell 210 and the (x′,y′) coordinate frame 224 of cell 220 is shown in FIG. 15.

Relative to coordinate frame 222, coordinate frame 224 is translated by vector (xt,yt), in this case (6,4), and scaled by a factor of four (and in general for an N×N tile, a factor of N).

When the tiling record for triangle 214 is created by procedure Transform & Set Up Polygon 900, step 908 computes coefficients (A,B,C) in the edge equation Ax+By+C=0 for edge 1502 relative to coordinate frame (x,y) of tile 210 (this is the smallest enclosing tile). When tile 210 is subdivided and a record for tile 220 is created, this edge equation is transformed to edge equation A′x′+B′y′+C′=0, which is relative to coordinate frame (x′,y′) of tile 220.

New coefficients (A′, B′, C′) are computed using the following transformation formulas 4000, which are applied to edge and plane equations at step 1342 of procedure 1300: A′=A/N B′=B/N C′=Axt+Byt+C.

Assuming that N is a power of two, A′ and B′ can be obtained by shifting. Frequently, Ax+By+C has already been evaluated at (xt, yt) at step 1308 or 1310 of procedure 1300, in which case C′ is already known. Whether or not this is exploited, C′ can be efficiently computed since xt and yt are small integers.

Thus, computing new coefficients for the line and plane equations is done very efficiently at step 1342 of procedure 1300, without performing general-purpose multiplication.

The same transformation formulas 4000 can be applied to any linear equation of the form w=Ax+By+C including edge equations, plane equations, and equations used in shading.

If shading is performed during hierarchical tiling at step 1332 of procedure 1300, the method can be applied to interpolating vertex colors of triangles (i.e., performing Gouraud shading). In this case, the intensities of the red, green, and blue color components can each be expressed as a linear equation (e.g. red=Ax+By+C) and evaluated in the same way as z-values'.

Since both sides of an equation can be multiplied by the same quantity, equation w=Ax+By+C is equivalent to equation Nw=N(Ax+By+C). Hence, using the following transformation formulas 4001 would result in computing Nw rather than w: A′=A B′=B C′=N(Axt+Byt+C).

In this case, coefficients A and B are unchanged but it is necessary to compute w from Nw by shifting (unless only the sign of the equation must be determined, as is the case when evaluating an edge equation).

Regardless of whether formulas 4000 or formulas 4001 are employed, transforming a linear equation from the coordinate frame of one tile to the coordinate frame of a “child” tile involves translation and scaling computations, where scaling is performed by shifting. With formulas 4000, scaling is performed by shifting coefficients A and B of the equation, and with formulas 4001, scaling is performed by shifting Axt+Byt+C, which is a linear expression of the coefficients of the equation.

This method for hierarchical evaluation of linear equations can also be applied in higher dimensions. For example, 3D tiling of a convex polyhedron into a voxel hierarchy having N×N×N decimation could be accelerated by hierarchical evaluation of the plane equations of the polyhedron's faces, which each have the form Ax+By+Cz+D=0. For cell (xt,yt,zt) within an N×N×N tile, the transformed coefficients of this equation are: A′=A/N B′=B/N C′=C/N D′=Axt+Byt+Czt+D, or equivalently, A′=A B′=B C′=C D′=N(Axt+Byt+Czt+D).

This method of hierarchical evaluation can be applied to evaluate higher-degree polynomial equations. For example, the general equation for a conic section (ellipse, parabola, or hyperbola) is Ax²+Bxy+Cy²+Dx+Ey+F=0. For cell (xt,yt) within an N×N tile, the transformed coefficients of this equation are: A′=A/N ² B′=B/N ² C′=C/N ² D′=(2Axt+Byt+D)/N E′=(2Cyt+Bxt+E)/N F′=Axt ² +Bxtyt+Cyt ² +Dxt+Eyt+F, or equivalently, A′=A B′=B C′=C D′=N(2Axt+Byt+D) E′=N(2Cyt+Bxt+E) F′=N ²(Axt ² +Bxtyt+Cyt ² +Dxt+Eyt+F).

Evaluation of these equations can be accelerated by computing some or all of the terms with shifting and addition, rather than multiplication. As with transforming linear equations, the above transformation formulas perform translation and scaling computations, and scaling is accomplished by shifting (shifting either a single coefficient or a polynomial expression of coefficients, such as expression 2Axt+Byt+D in the formula above).

The hierarchical evaluation methods described above can be applied when the image raster has jittered samples by scaling up the coordinate frame of the tiles. For example, if the coordinate frame of a 4×4 tile is scaled up by a factor of 4, there would be 32 integer values across the tile instead of 8, and the x and y coordinates of jittered image samples could have any of these values.

In summary, the hierarchical evaluation methods described above can be applied to accelerating processing of geometric objects described by polynomial equations within a spatial hierarchy (e.g., an image pyramid, octree, quadtree, etc.) that is organized in nested tiles that progress in scale by powers of two.

The method transforms a polynomial equation (e.g., a linear or quadratic equation of x and y) from the coordinate frame of one tile to the coordinate frame of a “child” tile at the adjacent finer level of the hierarchy. This transformation is performed by translation and scaling computations, where scaling is performed by shifting the binary representation of the equation's coefficients or by shifting the binary representation of a polynomial expression of the equation's coefficients.

Shifting can be used to scale numbers represented in floating-point format, in addition to numbers represented in integer format. The advantage of this method of hierarchical evaluation is that evaluation can often be done without performing general-purpose multiplication, thereby accelerating computation and simplifying the required circuitry.

Hierarchical evaluation of equations can be applied to a variety of tiling, shading, and interpolation computations which require evaluation of polynomial equations at samples within a spatial hierarchy. The method is well suited to implementation in hardware and it works well in combination with incremental methods.

Propagation of Z-Values.

While looping over cells within a finest-level tile, Process N×N Tile 1300 determines zfar_(x)[L] at each pyramid level L and the tile's zfar value (zfar_finest). Given this information, propagation can usually be performed with only one or two depth comparisons at each level of the pyramid (actually, this is only possible at levels where the ancestor_flag is TRUE, but this usually is the case).

The prior-art method of performing propagation during hierarchical z-buffering requires performing N² depth comparisons for N×N tiles at each level of propagation. The method described herein accelerates propagation by reordering most of these depth comparisons, performing them during tiling.

Another advantage of maintaining zfar_(x) values is that when propagation to an ancestor tile is not necessary, this can be determined without accessing z-values for the ancestor tile.

Suppose that ZFAR[F] is the farthest z-value within the current tile C in the finest level (where F is the index of the finest level), ZFAR[F−1] is the farthest z-value within the parent tile of the current tile, and so forth. Then the farthest z-values within ancestor tiles can be computed from zfar_finest and the values in array zfar_(x) as follows:

ZFAR[F]=zfar_finest (zfar within C),

ZFAR[F−1]=farthest of (ZFAR[F], zfar_(x)[F−1]) (zfar within parent of C),

ZFAR[F−2]=farthest of (ZFAR[F−1], zfar_(x)[F−2]) (zfar within grandparent of C),

and so forth.

Propagation can stop whenever it fails to change the existing value in an ancestor tile. The actual algorithm used to perform propagation will be presented after discussing procedure Update zfar 1600 (FIG. 16), which maintains array zfar.

Procedure Update zfar 1600 is called at step 1306 of Process N×N Tile 1300 to update zfar_(x) values. The procedure receives as input the index “I” of the current cell within the current tile.

Step 1602 initializes variable “L” to the finest level of the pyramid.

Next, at step 1604, if the z-pyramid cell with index I in z-array[L] (i.e., z-array[L][I]) covers the current tile, control proceeds to step 1610. Otherwise, at step 1606, if z-array[L][I] is farther than the current value of zfar_(x)[L], zfar_(x)[L] is set equal to z-array[L][I], and then control proceeds to step 1610.

At step 1610, if L is the coarsest level, the procedure terminates at step 1612. Otherwise, step 1614 sets L to the index of the adjacent coarser level and control returns to step 1604.

At any level L where ancestor_flag[L] is FALSE, zfar_(x)[L] is not a valid value and it will need to be recomputed later, but this is a relatively rare event. Although the method just described computes zfar_(x) values one by one, all values can be computed in parallel.

The propagation procedure, Propagate Z-Values 1700 (FIG. 17), is called after step 1120 of Tile Convex Polygon 1100. Step 1702 initializes variable L to the finest level of the pyramid and variable K to the next-to-finest level.

Next, if variable zfar_finest (zfar of the most recently processed finest-level tile) is not nearer than zfar_(x)[L], no propagation can be performed, so the procedure terminates at step 1706. Next, step 1708 sets variable zfar to variable zfar_finest.

Next, if ancestor_flag[K] is FALSE (step 1710), step 1712 reads the z-values corresponding to the level-K ancestor of the current cell from the z-pyramid into z-array[K] using procedure Read Z-Array 1200. If ancestor_flag[K] is TRUE at step 1710, control proceeds directly to step 1714.

Step 1714 determines the index “A” of the cell within array z -array[K] that is an ancestor of the z-value being propagated.

Next, step 1716 sets variable zold to the depth value for cell A in z-array[K] (i.e., z-array[K][A]).

Next, step 1718 overwrites z-array[K][A] with the value of variable zfar. Next, if K is the coarsest level (step 1720), step 1722 determines whether zfar is farther than zfar_(x)[K]. If so, zfar is a new zfar value for the entire z-pyramid, and step 1724 sets variable pyramid_zfar to variable zfar.

Whether or not step 1722 is executed, the procedure terminates at step 1726.

If K is not the coarsest level at step 1720, control proceeds to step 1728, where if Read Z-Array 1200 was executed at step 1712, zfar_(x)[K] is computed from the values in z-array[K] (this is a relatively slow procedure, but usually it is not required). Next, at step 1730, if zold is not farther than zfar_(x)[K], the procedure terminates at step 1732.

Otherwise, step 1734 sets variable zfar equal to the farthest of variables zfar and zfar_(x)[K]. Next, step 1736 sets L equal to K and sets K equal to the level that is adjacent to and coarser than L, and control returns to step 1710.

Although procedure Process N×N Tile 1300 updates array zfar_(x) while looping over individual cells in an N×N tile, the same approach could also be applied if several cells were computed in parallel, for example, if tiles were processed row-by-row instead of cell-by-cell.

When a new value of variable pyramid_zfar is established at step 1724, the far clipping planes maintained by the scene manager 110 and the z-buffer 180 can be reset to this nearer value.

Variable pyramid_zfar is part of the tip of the z-pyramid which is copied to the scene manager 110 at step 716 of procedure 700. The scene manager 110 uses pyramid_zfar to reset the far clipping plane, and it uses pyramid_zfar and other copied depth values to cull occluded bounding boxes, as described below.

Culling with the Tip of the Z-Pyramid.

When culling boxes with a z-pyramid, occlusion can sometimes be detected with a single depth comparison. However, when culling is performed with procedure Process Batch of Boxes 700, culling an occluded box requires transforming the box's front faces to perspective space, processing them with the culling stage 130, and reporting results to the scene manager 110.

To avoid the latency caused by these steps, an alternative is for the scene manager 110 to maintain some z-pyramid values and cull a box if it (or its bounding sphere) is occluded by a z-pyramid cell. Only if occlusion cannot be detected at this stage is a box sent through the rest of the system.

According to the method of the invention, after v-query results are reported to the scene manager 110 on the feedback connection 190 at step 714 of Process Batch of Boxes 700, step 716 copies the tip of the z-pyramid 170 to the scene manager 110. The “tip” includes the zfar value for the entire z-pyramid (i.e., pyramid_zfar), the coarsest N×N tile in the pyramid, and perhaps some additional levels of the pyramid (but not the entire pyramid, since this would involve too much work).

The amount of data that needs to be copied may be very modest. For example, if the copied tip includes pyramid_zfar, the coarsest 4×4 tile, and the 16 4×4 tiles at the adjacent finer level, a total of 273 z-values need to be copied. In some cases, the scene manager 110 can cull a substantial amount of occluded geometry using this relatively small amount of occlusion information.

At step 702 of procedure Process Batch of Boxes 700, the scene manager 110 uses the tip of the pyramid to perform conservative culling on occluded bounding boxes using procedure Is Box Occluded by Tip 1900 (FIG. 19). This culling procedure 1900 is illustrated in FIGS. 18 a and 18 b, which show the coordinate frame of model space 1800 (the coordinate frame that the model is represented in), bounding boxes 1802 and 1804, the view frustum 1806 with its far clipping plane 1810, the current zfar value of the z-pyramid (i.e., pyramid_zfar) 1812, and the current zfar values for a row 1814 of cells within the coarsest N×N tile 1816 of the z-pyramid 170, including the zfar value of cell 1820.

To simplify illustration, the frustum is oriented so that the viewing axis 1822 is parallel to the page and four faces of the frustum are perpendicular to the page.

If pyramid_zfar 1812 is nearer than the depth of the far clipping plane 1810, this establishes a nearer value for the far clipping plane, so the far clipping plane is reset to this value. In FIG. 18 a, resetting the far clipping plane to pyramid_zfar 1812 enables rapid culling of box 1802, since the depth of the nearest corner of box 1802 (which was computed at step 614 of procedure Sort Boxes into Layers 600) is farther than pyramid_zfar 1812.

Now the steps of procedure Is Box Occluded by Tip 1900 are described. The procedure is described infra as it applies to box 1804 in FIGS. 18 a and 18 b. Step 1902 determines whether the nearest corner of the box is farther than the far clipping plane.

If so, step 1912 reports that the box is occluded, and the procedure terminates at step 1916. If not, control proceeds to step 1904, which determines a bounding sphere 1824 for the box 1804, and step 1906 transforms the sphere's center 1826 to perspective space and determines the depth D 1828 of the sphere's nearest point.

Next, step 1908 determines the smallest z-pyramid cell 1820 that encloses the sphere 1824 and reads the cell's zfar value. If depth D 1828 is farther than zfar (step 1910), step 1912 reports that the box is occluded (this is the case with box 1804) and the procedure terminates at step 1916.

Otherwise, step 1914 reports that the box is potentially visible and the procedure terminates at step 1916.

Summarizing this culling method, the scene manager 110 receives the tip of the z-pyramid 170 along with v-query results on connection 190 and uses these z-values to reset the far clipping plane and perform conservative culling of bounding boxes. The method described supra for culling boxes with the tip of the z-pyramid is very efficient because processing a box only requires transforming a single point (or none) and making a single depth comparison.

The tip of the pyramid is in fact a low-resolution z-pyramid, that is, a z-pyramid with lower resolution than the z-pyramid 170 maintained by the culling stage 130, or if there is no separate culling stage, than the z-pyramid maintained by a hierarchical rendering stage.

Data Flow within the Culling Stage.

FIG. 20 shows a block diagram of data flow within the culling stage 130. This is a high-level schematic diagram that does not include all data and signals that would be required in an implementation.

The input to the culling stage 130 is the processing mode 2002, either render of v-query, and a list of records for transformed polygons 2004 sent by the geometric processor 120. First, data flow is described when the culling stage 130 is operating in render mode and rendering a list of polygons with Tile Polygon List 800.

In this case, the geometric processor 120 outputs two records for each polygon, a tiling record and a rendering record, and these records are buffered in the FIFO of Tiling Records 2006 and the FIFO of Rendering Records 2008, respectively.

Tile Polygon List 800 processes polygons one by one until all polygons on the list have been tiled. For each polygon, the Tile Stack 2010 is initialized by copying the next tiling record in the FIFO of Tiling Records 2006 on connection 2012 (step 1104). The Current Tile register 2014 is loaded from the Tile'Stack 2010 on connection 2016 (step 1108).

When Process N×N Tile 1300 performs occlusion and overlap tests (steps 1308 and 1310), edge and plane equations (which are part of tiling records) are read from the Current Tile register 2014 on connection 2022, and z-values are read from the list of z-arrays 2018 on connection 2024.

Whenever z-values are needed for a tile that is not stored in the list of z-arrays 2018, they are obtained from the z-pyramid 170, which involves writing an old tile record (if necessary) and reading a new tile record on connection 2020. When visible samples are encountered, z-values are written to the list of z-arrays 2018 on connection 2024 (step 1330). When z-values are propagated, z-values are read from and written to the list of z-arrays 2018 on connection 2024.

When new tiles are created (at step 1344), they are written to the Tile Stack 2010 on connection 2026.

When it is established that a polygon is visible (at step 1322 or step 1340), the polygon's record in the FIFO of Rendering Records 2008 is output to the z-buffer renderer 140 on connection 2028. Records in the FIFO of Rendering Records 2008 that correspond to occluded polygons are discarded.

Now data flow is considered when the culling stage 130 is operating in v-query mode and determining the visibility of bounding boxes with Process Batch of Boxes 700. In this case, the geometric processor 120 outputs tiling records and markers indicating “end of box” and “end of batch.” Tiling records are buffered in the FIFO of Tiling Records 2006. When in v-query mode, the geometric processor 120 does not output rendering records, so none are loaded into the FIFO of Rendering Records 2008.

Flow of tiling records on connections 2012, 2016, 2022, and 2026 is the same as when in rendering mode.

Z-values needed for depth comparisons at step 1308 are read from the list of z-arrays 2018 on connection 2024, but no z-values are written on this connection. If z-values are needed for a tile that is not stored in the list of z-arrays 2018, they are obtained from the z-pyramid 170, which involves writing an old tile record (if necessary) and reading a new tile record on connection 2020.

If a visible sample is discovered, the bit in V-Query Status Bits 2030 corresponding to the current box is set to visible on connection 2032 (step 710); otherwise the bit is set to occluded (step 712).

When the visibility of all boxes in the batch has been established, the V-Query Status Bits 2030 and the tip of the z-pyramid 170 are sent to the scene manager 110 on the feedback connection 190 (steps 714 and 716).

Other Ways of Reducing Image-Memory Traffic.

The culling stage preferably uses a low-precision z-pyramid 170 in order to reduce storage requirements and memory traffic.

The most straightforward way to implement a low-precision z-pyramid is to store each z-value in fewer bits than the customary precision of between 24 and 32 bits. For instance, storing z-values in 8 bits reduces storage requirements by a factor of 4 as compared with storing z-values in 32 bits.

Even greater reductions in the storage requirements of a z-pyramid used for conservative culling can be achieved with the modifications described below.

Encoding of Depth Values.

Storage requirements of the z-pyramid 170 can be reduced by storing depth information for tiles in a more compact form than N×N arrays of z-values.

According to this method, a finest-level tile is stored as a znear value and an array of offsets from znear, where znear is the depth of the nearest sample within the tile. Preferably, offsets are stored at relatively low precision (e.g., in 4 bits each) and znear is stored at higher precision (e.g., in 12 bits).

The record for each finest-level tile consists of an N×N array of offsets, znear, and a scale factor S that is needed to compute depths from offsets. If znear is stored in 12 bits, S in 4 bits, and each offset value in 4 bits, the record for a 4×4 tile requires 80 bits, which is 5 bits per sample. Z-values in tiles that are not at the finest level of the pyramid are stored in arrays, as usual (for example, as arrays of 8-bit z-values).

FIG. 21 shows a side view of a finest-level tile in the z-pyramid, which in three dimensions is a rectangular solid 2100 having a square cross-section. Given the indicated direction of view 2102, the right-hand end 2104 of the solid 2100 is the near clipping plane and the left-hand end 2106 of the solid 2100 is the far clipping plane.

The four thin horizontal lines 2116 indicate the positions of rows of samples within the tile. The two inclined lines, 2108 and 2110, indicate the positions of two polygons, which are oriented perpendicular to the page to simplify illustration.

In this instance, the depth of sample A on polygon 2110 is znear 2112, sample B is not “covered,” so its depth is the depth of the far clipping plane 2106, and sample C on polygon 2108 is the deepest covered sample within the tile. The depth of the deepest covered sample within a tile is called zfar_(c) (in this case, zfar_(c) is the depth 2114 of sample C).

To improve effective depth resolution, one offset value is reserved to indicate samples that lie at the far clipping plane (that is, samples that have never been covered by a polygon).

For example, suppose that offset values are each 4-bit values corresponding to integers 0 through 15, and value 15 is reserved to mean “at the far clipping plane.” Then, offset values 0 through 14 would be used to represent depths in the range znear to zfar_(c).

In general, this requires scaling by the scale factor S, computed with the following formula 6000: S=(FAR−NEAR)/(zfar_(c)−znear), where BEAR is the depth of the near clipping plane and FAR is the depth of the far clipping plane. Once S has been computed, the offset for a covered sample at depth z is computed with the following encoding formula 6001: offset=(z−znear)/S, where offset is rounded to an integer. The inverse decoding formula 6002 for computing a z-value from an offset is: z=znear S*offset.

To simplify evaluation of the encoding and decoding formulas, scale factor S is rounded to a power of two, which enables both multiplication and division by S to be performed by shifting. As a result, computations of both offsets and z-values are only approximate, but computations are structured so that depth comparisons are always conservative, never causing a visible polygon to be culled.

Given the previous assumptions about using 4-bit offsets, in FIG. 21, the offset computed for sample A would be 0 (because its depth is znear), the offset computed for sample C would be 14 (because its depth is zfar_(c)), the offset computed for sample D would lie somewhere between 0 and 14, and the offset for sample B would be 15, since this is the value reserved for “at the far clipping plane.”

znear and zfar_(c) can be computed by procedure Process N×N Tile 1300 as it loops over the cells within a tile. For example, to compute znear, step 1302 would initialize variable znear to the depth of the far clipping plane and following step 1326, variable znear would be updated with the depth of the nearest visible sample encountered so far.

When finest-level tiles in the z-pyramid are encoded, changes must be made when reading or writing a finest-level tile in procedures Tile Convex Polygon 1100 and Read Z-Array 1200: When Read Z-Array. 1200 reads the encoded record of a finest-level tile at step 1206, the z-value of each sample is computed from znear, S, and the offset value using the decoding formula 6002 and written to z-array[L] (where L is the finest level).

When writing the record for a finest-level tile, instead of writing z-array[L] at step 1120 of Tile Convex Polygon 1100, an encoded tile record is created from z-array[L] and then written to the z-pyramid 170. The tile record is created as follows.

First, the exponent of scale factor S is computed by computing S with formula 6000, rounding S to a power of two, and then determining its exponent (since S is a power of 2, it can be stored very compactly as an exponent).

Then the offset value corresponding to each z-value is computed. If S for the tile has not changed since the tile was read, the old offset is used for any sample where the polygon was not visible. Otherwise, the offset is computed using the encoding formula 6001.

Now all of the information in a tile record is known, and the record is written to the z-pyramid 170. Following step 1120 in Tile Convex Polygon 1100, propagation of the tile's zfar value is performed in the usual way using z-values that are not encoded.

The method described above makes it possible to construct highly accurate z-values from low-precision offsets whenever the depths of covered image samples within a tile lie within a narrow range, which is often the case. In the worst case when z-values cover nearly the whole range between the near and far clipping planes, this method is equivalent to representing z-values solely with low-precision offset values, compromising z-resolution. In typical scenes, however, depth coherence within finest-level tiles is quite high on average, resulting in accurate z-values and efficient culling in most regions of the screen.

Even though the finest level of the z-pyramid is not a conventional z-buffer when depth values are encoded as described above, herein the terms “z-pyramid” and “hierarchical depth buffer” will still be applied to this data structure.

Reducing Storage Requirements with Coverage Masks.

Another novel way to reduce the storage requirements of a z-pyramid used for conservative culling is to maintain a coverage mask at each finest-level tile and the zfar value of the corresponding samples, which together will be called a mask-zfar pair. According to this method, the record for each finest-level tile in the z-pyramid consists of the following information, which will be called a mask-zfar record 7000 for a tile.

Mask-Zfar Tile Record.

1. zfar value for the whole tile (zfar_(t))

2. mask indicating samples within a region of the tile (mask)

3. zfar value for the region indicated by mask_(t) (zfar_(m))

The terms zfar_(t), mask_(t) and zfar_(m) are defined above.

Preferably, only tiles at the finest level of the z-pyramid are stored in mask-zfar records. At all other levels, tile records are arrays of z-values which are maintained by propagation. Preferably, individual z-values within these arrays are stored at low precision (e.g., in 12 bits) in order to conserve storage.

The advantage of using mask-zfar records is that they require very little storage. For example, if zfar_(t) and zfar_(m) are each stored in 12 bits, the record for a 4×4-sample tile would require only 40 bits, 24 bits for these z-values and 16 bits for mask_(t) (one bit for each sample).

This is only 2.5 bits per sample, more than a three-fold reduction in storage compared with storing an 8-bit z-value for each sample, and more than a twelve-fold reduction in storage compared with storing a 32-bit z-value for each sample.

It is not essential to store zfar_(t) in mask-zfar records, because the identical z-value is also stored in the record for the parent tile. Eliminating zfar_(t) from mask-zfar records would reduce storage requirements to 1.75 bits per sample for a 4×4 tile, given the assumptions stated above. However, this approach requires that the parent tile's records be read more often when finest-level tiles are processed, which is a disadvantage.

FIG. 22 and FIG. 23 show an example illustrating how zfar_(t) advances when polygons that cover a tile are processed. FIG. 22 shows a 4×4 tile 2200 at the finest level of the z-pyramid having uniformly spaced samples 2202 that are covered by two triangles, labeled Q and R.

FIG. 23 shows a side view of the tile 2200, which in three dimensions is a rectangular solid 2300 having a square cross-section. Given the indicated direction of view 2302, the right-hand end 2304 of the solid 2300 is the near clipping plane and the left-hand end 2306 of the solid 2300 is the far clipping plane.

The four thin horizontal lines 2308 indicate the positions of rows of samples within the tile. The two inclined lines indicate the positions of triangles Q and R, which are oriented perpendicular to the page to simplify the illustration.

When the z-pyramid is initialized at the beginning of a frame, mask-zfar records in the z-pyramid are initialized as follows: zfar_(t) is set to the depth of the far clipping plane and mask_(t) is cleared to all zeros, meaning that no samples are covered. Thus, before processing any polygons at tile 2200, zfar_(t) is the depth of the far clipping plane 2306 and mask_(t) is all zeros.

Suppose that Q is the first polygon processed at tile 2200. When Q is processed, the bits in mask_(t) are set that correspond to the samples covered by Q (these are the samples within the crosshatched region 2204 in FIG. 22 b) and zfar_(m) is set to the depth of the farthest sample covered by Q, labeled zfar_(Q) in FIG. 23.

Later, when R is processed, its mask (indicated by the crosshatched region 2206 in FIG. 22 c) and its zfar value within the tile (labeled zfar_(R) in FIG. 23) are computed. Since R's mask 2206 and mask_(t) (in this case, Q's mask 2202) collectively cover the tile 2200, a nearer value has been established for zfar_(t), in this case zfar_(R), so zfar_(t) is set to zfar_(R).

This illustrates how zfar_(t) advances when one or more polygons covering a tile are processed, which enables conservative culling of occluded polygons that are encountered later.

Next, the general method is described for updating a mask-zfar record when a polygon is processed. Cases that need to be considered are schematically illustrated in FIG. 24.

FIG. 24 shows a side view of a 4×4 tile, which in three dimensions is a rectangular solid 2400 having a square cross-section. Given the indicated direction of view 2402, the right-hand end 2404 of the solid 2400 is the near clipping plane and the left-hand end 2406 of the solid 2400 is the far clipping plane.

The four thin horizontal lines 2408 indicate the positions of rows of samples within the tile. The bold vertical lines at depths zfar_(t) and zfar_(m) represent the occlusion information stored in the tile's mask-zfar record. The bold line at depth zfar_(t) covers the whole tile and the bold line at depth zfar_(m) indicates the samples covered by mask_(t).

The numeral 2410 identifies a polygon that is oriented perpendicular to the page.

The dashed vertical lines labeled P₁, P₂, P₃, P₄, and P₅ represent possible positions of the next polygon to be processed, indicating the region of the tile covered by visible samples on the polygon and the polygon's zfar value in relation to zfar_(m), and zfar_(t). Here, the “polygon's zfar value” is the farthest z of its potentially visible samples, so this z-value must be nearer than zfar_(t).

Although coverage is only depicted schematically, the basic cases are distinguished: the polygon covers the whole tile (case P₃), the polygon covers the tile in combination with mask_(t) (cases P₁ and P₄), and the polygon does not cover the tile in combination with mask_(t) (cases P₂ and P₅).

If each sample on a polygon lies behind zfar_(t) or is covered by mask_(t) and lies behind zfar_(m), the polygon is occluded within the tile. For example, polygon 2410 in FIG. 24 (oriented perpendicular to the page for convenience), is occluded because sample 2412 is inside mask_(t) and behind zfar_(m) and sample 2414 is behind zfar_(t).

When using mask-zfar records in the z-pyramid, changes must be made when reading or writing a finest-level tile in procedures Tile Convex Polygon 1100 and Read Z-Array 1200. When step 1206 of Read Z-Array 1200 reads the mask-zfar record of a finest-level tile (which includes zfar_(c), mask_(t) and zfar_(m)), the z-value of each sample is written to z-array[L] (where L is the finest level). The z-value of each sample covered by mask_(t) is zfar_(m) and the z-value of all other samples is zfar_(t).

When writing the record for a finest-level tile, instead of writing z-array[L] at step 1120 of Tile Convex Polygon 1100, a new mask-zfar record is created from z-array[L] with procedure Update Mask-Zfar Record 2500 and this record is written to the z-pyramid.

If all samples on the polygon are occluded (as with polygon 2410, for example), step 1120 is not executed, so neither is Update Mask-Zfar Record 2500.

Update Mask-Zfar Record 2500 (FIG. 25) receives as input the values in the old mask-zfar record (i.e., zfar_(t), zfar_(m), and mask_(t)), the mask for samples where the polygon is visible within the tile (call this mask_(p)), and the zfar value of these samples (call this zfar_(p)). mask_(p) and zfar_(p) can be computed efficiently within Process N×N Tile 1300 as it loops over the samples in a tile.

At step 2502, if mask_(p) covers the whole tile (i.e., it is all ones, which means that the polygon is visible at all samples, as for case P₃ in FIG. 24), at step 2504 zfar_(t) is set to zfar_(p) and mask_(t) is cleared to all zeros, and the procedure terminates at step 2506. Otherwise, control proceeds to step 2508 where if mask mask_(p) is all ones (where “|” is the logical “or” operation), the polygon and mask_(t) collectively cover the tile, and in this case, control proceeds to step 2510.

At step 2510, if zfar_(p) is nearer than zfar_(m) (e.g. P₄ in FIG. 24), a nearer zfar value has been established and step 2512 sets zfar_(t) to zfar_(m), mask_(t) to mask_(p), and zfar_(m) to zfar_(p), followed by termination at step 2514. If zfar_(p) is not nearer than zfar_(m) at step 2510 (e.g. P₁ in FIG. 24), step 2516 sets zfar_(t) to zfar_(p), followed by termination at step 2514.

If mask_(t)|mask_(p) is not all ones at step 2508, the polygon and mask_(t) do not collectively cover the tile, and the occlusion information for the polygon and mask_(t) are combined as follows. Step 2518 sets mask_(t) to mask_(t)|mask_(t) (where “|” is the logical “or” operation). Next, at step 2520, if mask_(t) is all zeros, control proceeds to step 2524, which sets zfar_(m) to zfar_(p), followed by termination of the procedure at step 2526.

If mask_(t) is not all zeros at step 2520, control proceeds to step 2522, where, if zfar_(p) is farther than zfar_(m), control proceeds to step 2524. For example, with P₂ in FIG. 24, zfar_(p) is farther than zfar_(m), so step 2524 would be executed.

If zfar_(p) is not farther than zfar_(m) at step 2522 (as is the case with P₅ in FIG. 24), the procedure terminates at step 2526.

Some of the operations performed by Update Mask-Zfar Record 2500 can be done in parallel.

In summary, the advantage of using mask-zfar pairs to store occlusion information in a z-pyramid used for conservative culling is that it requires very little storage (for example, 2.5 bits per image sample). The disadvantage of this approach is that maintaining occlusion information is more complicated and culling efficiency may not be as high.

To illustrate the savings in storage that can be achieved, when the finest level of a z-pyramid having 4×4 decimation is stored as mask-zfar tile records, each record including two 12-bit z-values and one 16-bit coverage mask_(t) and the other levels of the z-pyramid are stored as arrays of 12-bit z-values, the z-pyramid requires approximately 3.30 bits of storage per sample in the finest level. In this case, the total bits of storage in a 32-bit z-buffer having the same resolution is approximately ten times greater than the total bits of storage in the z-pyramid.

Even though the finest level of the z-pyramid is not a conventional z-buffer when mask-zfar records are employed, herein the terms “z-pyramid” and “hierarchical depth buffer” will still be applied to this data structure.

The prior art includes the A-buffer visible-surface algorithm that maintains pixel records that include coverage masks and z-values. At individual pixels, the A-buffer algorithm maintains a linked list of visible polygon fragments, the record for each fragment including a coverage mask indicating the image samples covered by the fragment, color and opacity values, and znear and zfar values, each stored in floating-point format. This record format is designed to resolve color and visibility at each image sample, enabling high-quality antialiasing of pixel values.

Although the A-buffer record format could be employed at finest-level tiles in the z-pyramid, its variable-length, linked-list format greatly complicates processing and requires dynamic memory allocation. By comparison, the novel method of performing conservative occlusion culling using a single coverage mask at a tile is much simpler and much easier to implement in hardware.

Culling with a Low-Resolution Z-Pyramid.

As previously mentioned, a separate culling stage 130 in the graphics system 100 enables conservative culling with a low-precision z-pyramid, that is, a z-pyramid having the same resolution as the z-buffer, but in, which z-values are stored at low precision, for example, as 8-bit or 12-bit values. Alternatively, the culling stage 130 can employ a low-resolution z-pyramid, that is, a z-pyramid having lower resolution than the z-buffer. As previously mentioned, the resolution of a z-pyramid is the resolution of its finest level.

For example, a single zfar value could be maintained in the finest level of the z-pyramid for each 4×4 tile of image samples in the output image 150. As applied to the 64×64 image raster of FIG. 2 (only partially shown), level 230 would be the finest level of the low-resolution z-pyramid, and each cell within this level would represent a conservative zfar value for the corresponding 4×4 tile of image samples in the image raster. For instance, cell 218 would contain a conservative zfar value for the image samples in 4×4 tile 220.

Definitive visibility tests cannot be performed using a low-resolution z-pyramid, but conservative culling can be performed. The disadvantage of a low-resolution z-pyramid is that it has lower culling efficiency than a standard z-pyramid, and this increases the workload on the z-buffer renderer 140.

However, a low-resolution z-pyramid has the advantage of requiring only a fraction of the storage, and storage requirements can be further reduced by storing zfar values at low-precision (e.g., 12 bits per value). In cases where the reduction in storage requirements enables the z-pyramid to be stored entirely on-chip, the resulting acceleration of memory access can improve performance substantially. In short, using a low-resolution z-pyramid impairs culling efficiency but reduces storage requirements and can increase culling speed in some cases.

To illustrate the savings in storage that can be achieved with a low-resolution z-pyramid, consider a graphics system with a 1024 by 1024 z-buffer in the rendering stage and a 256 by 256 z-pyramid in the culling stage. Assuming 32-bit z-values in the z-buffer, 12-bit z-values in the z-pyramid, and 4×4 decimation from level to level of the z-pyramid, the total bits of storage in the z-buffer would be approximately 40 times greater than the total bits of storage in the z-pyramid.

Using a low-resolution z-pyramid requires only minor changes to the rendering algorithm that has already been described for the graphics system 100 of FIG. 1. In fact, it is only necessary to change procedure Process N×N Tile 1300.

At step 1324, control proceeds to step 1334, which determines whether the polygon completely “covers” the cell. This occurs only if the cell is completely inside all of the polygon's edges. Whether a cell lies completely inside an edge can be determined with the edge-cell test described in connection with step 1310, except that instead of substituting the cell's corner that is farthest in the “inside direction” into the edge equation, the opposite corner is substituted.

If the polygon does not completely cover the cell, control returns to step 1304. Otherwise, step 1336 computes the zfar value of the plane of the polygon within the cell, which is done as previously described for computing the plane's znear value at step 1308, but instead of substituting the “nearest corner” of the cell into the plane equation, the opposite corner is substituted, since this is where the plane is farthest within the cell.

In FIG. 14, for example, the corner 1408 is the “nearest corner” of cell 1402, meaning that the plane of polygon 1400 is nearest to the observer at that corner. Therefore, the plane of polygon 1400 is farthest from the observer at the opposite corner 1410, so to establish the zfar value for the plane of polygon 1400 within cell 1402, the x and y coordinates of this corner 1410 are substituted into the plane equation, which has the form z=Ax+By+C.

If at step 1336 the plane's zfar value is nearer than the corresponding value for the current cell in z-array[F] (where F is the index of the finest level), control proceeds to step 1326, which sets changed to TRUE. Then step 1328 updates zfar_finest, overwriting zfar_finest with the plane's zfar value, if the plane's zfar value is farther than the current value of zfar_finest. Next, step 1330 overwrites the value for the current cell in z-array[F] with the plane's zfar value, and control returns to step 1304.

The optional shading step 1332 is not compatible with using a low-resolution z-pyramid. At step 1336, if the plane's zfar value is not nearer than the corresponding value in z-array[F], control returns directly to step 1304.

FIG. 26 shows a side view of a cell in the z-pyramid, which in three dimensions is a rectangular solid 2600 having a square cross-section. Given the indicated direction of view 2602, the right-hand end 2604 of the solid 2600 is the near clipping plane and the left-hand end 2606 of the solid 2600 is the far clipping plane. The bold vertical line indicates the current z-value 2608 stored in the z-pyramid cell.

The three inclined lines, 2610, 2620, and 2630, indicate the positions of three polygons, each covering the cell and each oriented perpendicular to the page to simplify illustration. For each polygon, its znear and zfar values within the cell are shown by dashed lines.

Now, the procedure Process N×N Tile 1300 processes these polygons within this cell, assuming a low-resolution z-pyramid.

Polygon 2610 would be determined to be occluded within the cell at step 1308, because its znear value 2612 is farther than the z-pyramid value 2608.

Polygon 2620 would be determined to be visible because its znear value 2622 is nearer than the current z-pyramid value 2608, but the z-pyramid would not be overwritten with the polygon's zfar value 2624 because the polygon's zfar value 2624 is farther than the current z-pyramid value 2608.

Polygon 2630 would be determined to be visible because its znear value 2632 is nearer than the current z-pyramid value 2608, and the z-pyramid would be overwritten with the polygon's zfar value 2634 because the polygon's zfar value 2634 is nearer than the current z-pyramid value 2608.

Now an alternative way of updating a low-resolution z-pyramid in the culling stage 130 is described. When the z-buffer renderer 140 encounters visible depth samples on a polygon, they are copied to the culling stage 130 and propagated through the z-pyramid 170.

This method requires a connection 185 for copying z-values from the z-buffer renderer 140 to the culling stage 130, which is drawn in a dashed arrow in FIG. 1 to indicate that this is just an option. If z-values in the z-pyramid 170 are stored at lower precision than z-values in the z-buffer 180, z-values may be converted to low-precision values before they are copied. When the culling stage 130 receives new depth samples on connection 185, they are propagated through the z-pyramid using the traditional propagation algorithm.

When this method is employed, it is not necessary to update the z-pyramid during tiling of polygons by the culling stage, which simplifies the tiling algorithm considerably. In fact, in procedures Tile Convex Polygon 1100 and Process N×N Tile 1300, only the steps performed in v-query mode are necessary, except for outputting rendering records when visible polygons are encountered.

Varying Z Precision within a Z-Pyramid.

In the description of procedure Process N×N Tile 1300, for the preferred embodiment of the invention, the culling and rendering stages are separate and have their own depth buffers, but it is possible to combine the two stages in a single “hierarchical renderer” having a single z-pyramid used for both culling and rendering.

In this case, the finest level of the z-pyramid is a z-buffer in which z-values are stored at full precision (e.g., in 32 bits per z-value) so that visibility can be established definitively at each image sample. At other pyramid levels, however, it is not necessary to store z-values at full precision, since culling at those levels is conservative.

Thus, at all but the finest pyramid level, it makes sense to store z-values at low precision (e.g., in 12 bits) in order to conserve storage and memory bandwidth and improve caching efficiency. Frequently, only z-values at coarse levels of the pyramid need to accessed to determine that a bounding box or primitive is occluded, so caching the coarsest levels of the pyramid can accelerate culling significantly. Using low-precision z-values enables more values to be stored in a cache of a given size, thereby accelerating culling.

When low-precision z-values are employed in a pyramid as described above, the average precision of z-values in the z-buffer is higher than the average precision of z-values in the entire z-pyramid. For example, for a z-pyramid having 4×4 decimation from level to level and a 1024 by 1024 z-buffer in which z-values are stored at 32 bits of precision, and in which z-values in the other pyramid levels are stored at 12 bits of precision, then the average z-precision in the z-buffer is 32 bits per z-value and average z-precision in the entire z-pyramid is approximately 30.9 bits per z-value.

Exploiting Frame Coherence.

As described supra, the efficiency of hierarchical z-buffering with box culling is highly sensitive to the order in which boxes are traversed, with traversal in near-to-far occlusion order achieving maximal efficiency. Render Frames with Box Culling 500 achieves favorable traversal order by explicitly sorting boxes into “layers” every frame.

Another method for achieving efficient traversal order, which is described next, is based on the principle that bounding boxes that were visible in the last frame are likely to be visible in the current frame and should, therefore, be processed first.

This principle underlies the procedure, Render Frames Using Coherence 2700 (FIG. 27), which works as follows. The scene manager 110 maintains four lists of box records:

1. boxes that were visible last frame (visible-box list 1);

2. boxes that were not visible last frame (hidden-box list 1);

3. boxes that are visible in the current frame (visible-box list 2); and

4. boxes that are not visible in the current frame (hidden-box list 2).

“Hidden” boxes include both occluded and off-screen boxes. In step 2702, the scene manager 110 organizes all scene polygons into polyhedral bounding boxes, each containing some manageable number of polygons (e.g., between 50 and 100). In step 2704, the scene manager 110 clears visible-box list 1 and hidden-box list 1, and appends all boxes in the scene to hidden-box list 1.

Now the system has been initialized and is ready to render sequential frames. First, step 2706 initializes the output image 150, z-pyramid 170, and z-buffer 180 (z-values are initialized to the depth of the far clipping plane).

Next, step 2708 reads boxes in first-to-last order from visible-box list 1 and processes each box, as follows. First, it tests the box to see if it is outside the view frustum, and if the box is outside, its record in the list is marked off-screen.

If the box is not outside, the polygons on its polygon list are rendered with procedure Render Polygon List 300. When the first frame is rendered, visible-box list 1 is null, so step 2708 is a null operation.

Next, step 2710 reads boxes in first-to-last order from hidden-box list 1 and processes each box as follows. First, it tests the box to see if it is outside the view frustum, using the method described at step 608 of procedure 600, and if the box is outside, its record in the list is marked off-screen.

If the box is not outside and it intersects the “near face” of the view frustum, its record in the list is marked visible and its polygons are rendered with Render Polygon List 300. If the box is not outside and it does not intersect the near face, it is tested for occlusion with respect to the tip of the z-pyramid with Is Box Occluded by Tip 1900, and if it is occluded, its record in the list is marked occluded.

Otherwise, the box is batched together with other boxes (neighbors on hidden-box list 1) and processed with Process Batch of Boxes 700 operating in render mode. If the box is visible, this procedure 700 renders the box's polygon list. Otherwise, the box's record in the list is marked occluded.

Now all polygons in visible boxes have been rendered into the output image 150, which is displayed at step 2712. The remaining task before moving on to the next frame is to establish which boxes are visible with respect to the z-pyramid.

First, step 2714 clears visible-box list 2 and hidden-box list 2. Next, step 2716 reads boxes in first-to-last order from visible-box list 1 and processes each box as follows. If the box was marked off-screen at step 2708, it is appended to hidden-box list 2. If the box was not marked off-screen and it intersects the “near face” of the view frustum, the box is appended to visible-box list 2.

If the box was not marked off-screen and it does not intersect the near face, it is tested for occlusion with respect to the tip of the z-pyramid with Is Box Occluded by Tip 1900, and if it is occluded, the box is appended to hidden-box list 2. Otherwise, the box is batched together with other boxes (neighbors on visible-box list 1) and processed with Process Batch of Boxes 700 operating in v-query mode in order to determine its visibility. If the box is visible, it is appended to visible-box list 2, and if it is occluded, it is appended to hidden-box list 2.

Next, step 2718 reads boxes in first-to-last order from hidden-box list 1 and processes each box as follows. If the box was marked off-screen or occluded at step 2710, it is appended to hidden-box list 2. If the box was marked visible at step 2710, it is appended to visible-box list 2.

Otherwise, the box is batched together with other boxes (neighbors on hidden-box list 1) and processed with Process Batch of Boxes 700 operating in v-query mode in order to determine its visibility. If the box is visible, it is appended to visible-box list 2, and if it is occluded, it is appended to hidden-box list 2.

Next, step 2720 renames hidden-box list 2 to hidden-box list 1 and renames visible-box list 2 to visible-box list 1. Then, step 2722 updates the bounds of boxes containing moving polygons (if any), and control returns to step 2706 to begin the next frame.

When there is a high degree of frame coherence, as is usually the case with animation, after rendering the first frame, the algorithm just described approaches the efficiency of near-to-far traversal while avoiding the trouble and expense of performing explicit depth sorting or maintaining the scene model in a spatial hierarchy. Efficient traversal order results from processing boxes first that were visible in the preceding frame (i.e., the boxes on visible-box list 1).

In addition, the order of boxes on the lists is the order in which their visibility was established, which is often correlated with occlusion order, particularly if the viewpoint is moving forward. Consequently, first-to-last traversal of lists improves the culling efficiency of procedure Render Frames Using Coherence 2700.

A similar strategy for exploiting frame coherence has been employed to accelerate z-buffering of models organized in an octree when the z-pyramid is maintained in software and cannot be accessed quickly by the polygon-tiling hardware.

Tiling Look-Ahead Frames to Reduce Latency

When rendering complex scenes in real time, the amount of storage needed for a scene model may exceed the capacity of memory that is directly accessible from the scene manager 110, called scene-manager memory. In this case, it may be necessary during the rendering of a frame to read part of the scene model from another storage device (e.g., a disk), which causes delay. Such copying of scene-model data into scene-manager memory from another storage device will be referred to as paging the scene model.

Paging the scene model can be controlled with standard virtual-memory techniques, “swapping out” data that has not been recently accessed, when necessary, and “swapping in” data that is needed.

When rendering scene models that are too large to fit in scene-manager memory, preferably, frames are rendered with procedure Render Frames with Box Culling 500 and records for bounding boxes are stored separately from the list of primitives that they contain. The record for a bounding box includes records for its faces and a pointer to the list of primitives that the box contains. Box records are retained in scene-manager memory and the lists of polygons associated with bounding boxes are swapped into and out of scene-manager memory as necessary.

The advantage of organizing the scene model in this way is that the only time that paging of the scene model is required when rendering a frame is when a bounding box is visible and its polygon list is currently swapped out. This occurs, for example, at step 720 of procedure Process Batch of Boxes 700, if the polygon list associated with a visible bounding box is not already present in scene-manager memory, in which case the polygon list must be copied into scene-manager memory before the scene manager 110 can initiate rendering of the polygon list.

Although the approach just described can reduce paging of the scene model, at some frames a large number of bounding boxes can come into view, and when this occurs, the time it takes to copy swapped-out lists of polygons into scene-manager memory can delay rendering of the frame.

The “look-ahead” method employed herein to reduce such delays is to anticipate which bounding boxes are likely to come into view and read their polygon lists into scene-manager memory, if necessary, so they will be available when needed. This approach enables delays caused by paging of the model to be distributed over a sequence of frames, resulting in smoother animation.

According to this method, first it is estimated where the view frustum will be after the next few frames have been rendered. This estimated frustum will be called the look-ahead frustum.

Then, as the next few frames are being rendered, a look-ahead frame corresponding to the look-ahead frustum is created using a procedure that is similar to rendering an ordinary frame, except that no output image is produced. Rather, processing of primitives stops after they are tiled into a z-pyramid, which is separate from the z-pyramid used to render ordinary frames and which will be called the look-ahead z-pyramid.

When tiling of a look-ahead frame has been completed all primitives which are visible in that frame have been paged into scene-manager memory and will be available if they are needed when rendering ordinary frames.

To support creation of look-ahead frames in the graphics system of FIG. 1, the culling stage 130 includes a look-ahead z-pyramid 195 (shown in dashed lines to indicate that this is just an option) and frame-generation procedures are modified so that a look-ahead frame can be generated gradually while one or more ordinary frames are being rendered.

Look-ahead frames are created with procedure Create Look-Ahead Frame 2900, shown in FIG. 29. This procedure is similar to rendering an ordinary frame with box culling, except that primitives are not passed on to the z-buffer renderer 140 after they are tiled into the look-ahead z-pyramid 195. This procedure 2900 is executed a little at a time, as the graphics system renders ordinary frames.

Procedure Create Look-Ahead Frame 2900 begins with step 2902, which clears the look-ahead z-pyramid 195 to the far clipping plane.

Next, step 2904 estimates where the view frustum will be after some small amount of time, for example, where the view frustum will be after another twenty frames have been rendered. This look-ahead frustum is determined by extrapolating the position of the viewpoint based on the position of the viewpoint in preceding frames, extrapolating the direction of view based on the direction of view in preceding frames, and constructing a frustum from the extrapolated viewpoint and direction of view. Preferably, look ahead frames are created with a wider view angle than ordinary frames so that more of the scene will be visible.

Next, procedure Sort Boxes into Layers 600, which has already been described, sorts the scene model's bounding boxes into layer lists to facilitate their traversal in approximately near-to-far order within the look-ahead frustum. This procedure also creates a near-box list containing the boxes that intersect the near face of the look-ahead frustum. To distinguish these lists from the lists used when rendering ordinary frames, they will be called the look-ahead layer lists and the look-ahead near-box list.

Next, step 2906 processes the polygon lists associated with the bounding boxes on the look-ahead near-box list. First, any of these polygon lists which are not already present in scene-manager memory are copied into scene-manager memory. Then, each polygon list is tiled into the look-ahead z-pyramid 195 using a modified version of procedure Render Polygon List 300 which operates as previously described, except that the procedure and its subprocedures access the look-ahead z-pyramid 195 (instead of the other z-pyramid 170), procedure Transform & Set Up Polygon 900 does not create or output rendering records, procedure Process N×N Tile 1300 does not output polygons to the z-buffer renderer 140, and step 306 of Render Polygon List 300 is omitted.

Next, step 2908 processes the look-ahead layer lists using a modified version of procedure Process Batch of Boxes 700. This procedure 700 operates as previously described, except that it and its subprocedures access the look-ahead z-pyramid 195 (instead of the other z-pyramid 170) and procedure Render Polygon List 300 (executed at step 720) is modified as described above.

To enable step 702 of procedure. Process Batch, of Boxes 700 to cull bounding boxes that are occluded by the look-ahead z-pyramid 195, the culling stage copies the tip of the look-ahead z-pyramid 195 to the scene manager 110 at step 716 of procedure Process Batch of Boxes 700. The scene manager 110 stores this occlusion data separately from the tip of the other z-pyramid 170.

At step 720 of procedure Process Batch of Boxes 700, if a polygon list is not already present in scene-manager memory, it must be copied into scene-manager memory prior to tiling with procedure Render Polygon List 300.

Following step 2908, procedure Create Look-Ahead Frame 2900 terminates at step 2910, and work begins on the next look-ahead frame. When a look-head frame is completed, all polygons which are visible in that frame have been copied into scene-manager memory and will be available if they are needed when rendering an ordinary frame.

Execution of procedure Create Look-Ahead Frame 2900 is interleaved with execution of steps 504 through 510 of procedure Render Frames with Box Culling 500 (which renders ordinary frames), with the scene manager 110 controlling switching from one procedure to the other.

Preferably, work on look-ahead frames is done at times when the components that it requires are not being used by Render Frames with Box Culling 500. For example, when Process Batch of Boxes 700 is rendering an ordinary frame, after a batch of bounding boxes is processed by the geometric processor 120 and the culling stage 130, there is a delay before the associated polygon lists are sent through the system, since it takes time to report the visibility of boxes. During this delay, a batch of boxes for the look-ahead frame can be processed by the geometric processor 120 and the culling stage 130.

Also, if processing of an ordinary frame is completed in less than the allotted frame time (e.g., in less than one thirtieth of a second), work can be performed on a look-ahead frame.

Preferably, the resolution of the look-ahead z-pyramid 195 is lower than the resolution of ordinary frames in order to reduce storage requirements, computation, and memory traffic. For example, the look-ahead z-pyramid 195 could have a resolution of 256×256 samples.

Preferably, even when a low-resolution look-ahead z-pyramid 195 is employed, the “ordinary” tiling algorithm is employed within procedure Process N×N Tile 1300, where control passes from step 1316 to step 1326, rather than step 1334 (step 1322 is skipped when tiling a look-ahead frame). In other words, steps 1334 and 1336 are only executed when tiling an ordinary frame with a low-resolution z-pyramid, not when tiling a look-ahead frame with a low-resolution z-pyramid.

Preferably, the look-ahead z-pyramid 195 is low-precision in addition to being low-resolution, in order to reduce storage requirements and memory traffic. For example, each z-value can be stored as a 12-bit value. Storage requirements can be further reduced by storing finest-level N×N tiles in the look-ahead z-pyramid 195 as mask-zfar pairs.

Hierarchical Z-Buffering with Non-Conservative Culling

Even with the efficiency of hierarchical z-buffering, at some level of complexity it may not be possible to render a scene within the desired frame time. When this occurs, accuracy can be traded off for speed by culling objects that may be slightly visible, that is, by performing non-conservative occlusion culling. Although this can noticeably impair image quality, in some cases this is acceptable for faster frame generation.

The speed versus accuracy tradeoff is controlled as follows. The error limit is defined as the maximum number of tiling errors that are permitted within a finest-level tile of the z-pyramid when tiling a particular polygon. A tiling error consists of failing to overwrite an image sample where a polygon is visible.

Using an error limit E permits non-conservative culling to be performed with one modification to the basic algorithm for hierarchical tiling. When propagating depth values through the z-pyramid, at each finest-level tile, instead of propagating the farthest z-value to its parent tile, the z-value of rank E is propagated, where the farthest z-value has rank 0 and the nearest z-value has rank N²−1.

Thus, when E is 0 the farthest z is propagated, when E is 1 the next-to-the-farthest z is propagated, when E is 2 the next-to-the-next-to-the-farthest z is propagated, and so forth. When propagating at other levels of the pyramid (i.e., except when propagating from the finest level to the next-to-the-finest level), the farthest z value in the child tile is propagated, as in a traditional z-pyramid.

Using this propagation procedure, except at the finest level, each z-value in the z-pyramid is the farthest rank-E z-value for any finest-level tile in the corresponding region of the screen. It follows that the occlusion test performed at step 1308 of procedure Process N×N Tile 1300 will automatically cull a polygon in any region of the screen where it is potentially visible at E or fewer image samples within any finest-level tile.

This method avoids some of the subdivision required to definitively establish the visibility of polygons or portions of polygons that are potentially visible at only a small number of image samples, thereby reducing both memory traffic and computation. Moreover, this advantage is compounded when culling bounding boxes, since culling of a “slightly visible” box saves the work required to process all polygons inside it.

Each polygon which is potentially visible at more than E image samples within a finest-level tile is processed in the usual way, so all of its visible image samples within these tiles are written.

This method of non-conservative culling requires the following modifications to procedure Process N×N Tile 1300, assuming an error limit of E.

First, instead of maintaining the farthest of the existing z-values for a finest-level tile in variable zfar_(x)[F] (where F is the index of the finest pyramid level), the z-value of rank E among the existing z-values for that tile is maintained. For example, if E is one, after looping over a finest-level tile in procedure Process N×N Tile 1300, variable zfar_(x)[F] contains the next-to-the-farthest z-value of the z-values that were originally stored for that tile. This modification requires changing procedure Update zfar_(x) 1600 when variable L is the index of the finest level.

Second, instead of maintaining the farthest z-value encountered so far for the tile being processed in variable zfar_finest, the z-value of rank E among those z-values is maintained in zfar_finest. For example, if E is one, after looping over a finest-level tile in procedure Process N×N Tile 1300, variable zfar_finest would contain the next-to-the-farthest z-value in z-array[F], where F is the index of the finest pyramid level.

Given these two modifications, procedure Propagate Z-Values 1700 propagates the correct z-values through the z-pyramid.

One way of thinking of this method for non-conservative occlusion culling is that the error limit provides a convenient, predictable “quality knob” that controls the speed versus quality tradeoff. When the error limit is zero, the method performs standard hierarchical z-buffering and it produces a standard image that is free of visibility errors. Otherwise, the higher the error limit, the faster the frame rate but the poorer the image quality.

When it is important to maintain a particular frame rate, the error limit can be adjusted accordingly, either by the user or by the rendering program, either at the beginning of a frame or during frame generation.

The method can be applied whether the image is point sampled or oversampled, so the speed versus quality spectrum ranges from relatively fast generation of point-sampled images with numerous visibility errors to relatively slow generation of accurately antialiased images that are free of visibility errors.

One shortcoming of this method of non-conservative culling is that it is possible that up to E image samples may never be tiled within a finest-level tile, even though they are covered by polygons that have been processed. This behavior can be avoided by adding an additional propagation rule: always propagate the farthest z-value until all image samples within a finest-level tile have been covered.

Other simple modifications to propagation rules may also improve image quality. For example, to make errors less noticeable propagation rules could be structured to avoid errors at adjacent image samples.

If multiple depth values are maintained corresponding to multiple error limits in the z-pyramid, different error limits can be selected depending on circumstances. For example, a higher error limit could be used when tiling bounding boxes than when tiling primitives, since culling a bounding box can save a lot of work. This approach does not require any changes to the finest level of the z-pyramid, but it requires propagating and storing multiple z-values for each cell at the coarser levels of the z-pyramid.

For example, if two z-values are maintained for each child tile at cells in levels of the z-pyramid that are coarser than the finest level, the farthest z-value and the next-to-the-farthest z-value within the corresponding region of the screen, then the farthest z-values could be applied to culling primitives and the next-to-the-farthest z-values could be applied to culling bounding boxes.

Summarizing the changes to the z-pyramid that are required when performing non-conservative culling for an error limit of E, the same information is stored at the finest level as with ordinary conservative culling, but at all coarser levels, instead of storing the farthest z-value within the corresponding region of the screen, the rank-E z-value for the corresponding region of the screen is stored. For example, if E is one, each z-value at levels that are coarser than the finest level is the next-to-the-farthest z-value for the corresponding region of the screen.

To support culling with K different error limits, it is necessary to store K z-values for each z-pyramid cell at levels of the pyramid that are coarser than the finest level, each of these K z-values corresponding to one of the error limits.

Implementation Issues.

Although each of the stages in the graphics system 100 of FIG. 1 can be implemented in either software or hardware, at the present time, it is more practical to implement the scene manager 110 in software and to implement the culling stage 130 and the z-buffer renderer 140 in hardware. Software implementation of the scene manager 110 is preferred because of the relative complexity of the operations it performs and the flexibility that software implementation provides.

Hardware implementation of the culling stage 130 and the z-buffer renderer 140 is preferred because, presently, it is not practical to attain real-time rendering of very complex scenes with software implementations. Although operations of the geometric processor 120 can be accelerated by hardware implementation, a software implementation running on the host processor (or another general-purpose processor) may provide adequate performance.

As processor performance improves over time, implementation of the entire system in software running on one or more general-purpose processors becomes increasingly practical.

Effectiveness of the Present Method of Occlusion Culling.

The graphics system 100 of FIG. 1 was simulated to compare its efficiency to traditional z-buffer systems when processing densely occluded scenes. The simulation employed a building model which was constructed by replicating a polygonal model of an office cubicle. By varying the amount of replication, scenes were created with depth complexities ranging from 3 to 53. These scene models are poorly suited to culling using the “rooms and portals” method because of their relatively open geometry.

A simulation program measured traffic on two classic bottlenecks in z-buffer systems: the traffic in polygons that need to be processed by the system, which will be referred to as geometry traffic, and depth-buffer memory traffic generated by depth comparisons, which will be called z-traffic and is measured in average number of bits of memory traffic per image sample. In the graphics system 100 of FIG. 1, geometry traffic is the traffic in polygons and cube faces on connections 115 and 125 and z-traffic is the combined traffic on connections 165 and 175.

Simulations compared z-buffering to hierarchical z-buffering, with and without box culling. The figures cited below assume that within the graphics system 100 the z-buffer 180 and output image 150 have resolution 1024 by 1024 and the z-pyramid 170 has resolution 1024 by 1024 and is organized in five levels of 4×4 tiles which are accessed on a tile-by-tile basis. This system is compared to a conventional z-buffer system with a 1024 by 1024 z-buffer having 32-bit z-values which are accessed in 4×4 tiles.

When processing versions of the scene having high depth complexity, the amount of geometry traffic was very high when box culling was not employed. For example, in a version of the scene with a depth complexity of 53, there were approximately 9.2 million polygons in the view frustum, and without box culling it was necessary to process all of these polygons every frame. With box culling and near-to-far traversal, it was only necessary to process approximately 45,000 polygons per frame, approximately a 200-fold reduction.

The advantage of hierarchical z-buffering over conventional z-buffering is that it reduces z-traffic dramatically, assuming favorable traversal order. For example, with box culling and near-to-far traversal of bounding boxes, for a version of the scene with a depth complexity of 16, z-buffering generated approximately 10 times as much z-traffic as hierarchical z-buffering using a z-pyramid with 8-bit z-values.

When scene depth complexity was increased to 53, z-buffering generated approximately 70 times as much z-traffic as hierarchical z-buffering using a z-pyramid with 8-bit z-values. For these scenes, performing box culling with conventional z-buffering was not effective at reducing z-traffic because boxes overlapped very deeply on the screen and the culling of occluded boxes generated a great deal of z-traffic.

The relative advantage of hierarchical z-buffering was less when traversal order was less favorable, but even when scene geometry was traversed in random order, hierarchical z-buffering generated substantially less z-traffic than traditional z-buffering.

Even without box culling, hierarchical z-buffering reduced z-traffic substantially. For example, when a version of the scene having a depth complexity of 16 was rendered without box culling, z-buffering generated approximately 7 times as much z-traffic as hierarchical z-buffering.

Next, culling performance was measured when finest-level tiles in the z-pyramid were stored as mask-zfar pairs with 12-bit zfar values. Tiles at coarser levels of the z-pyramid were stored as arrays of 12-bit z-values.

Compared to using a z-pyramid in which all tiles were stored in arrays of 8-bit z-values, this method improved culling efficiency, thereby reducing geometry traffic, and reduced z-traffic by a factor of three or four. Overall, a z-pyramid in which finest-level tiles are represented as mask-zfar pairs and tiles at coarser levels are represented as arrays of low-precision z-values appears to produce the best performance.

While the invention has been described with substantial particularity and has been shown with reference to preferred forms, or embodiments, it will be understood by those skilled in this art that other changes, than those mentioned, can be made. Therefore, it is understood that the scope of the invention is that defined by the appended claims. 

1. In a graphics system for processing geometric objects, said graphics system including a spatial hierarchy having a plurality of levels ranging from a finest level to a coarsest level, said spatial hierarchy being organized as a plurality of nested tiles that progress in scale by powers of two, each of said tiles defining a coordinate frame, said tiles including a first tile and a second tile, the second tile being nested inside the first tile, a method for evaluating a polynomial equation having coefficients and describing a geometric object, said method comprising: computing said coefficients of said polynomial equation relative to said coordinate frame of said first tile; and transforming said polynomial equation to said coordinate frame of said second tile.
 2. A method as defined in claim 1 wherein said polynomial equation is for a conic section and has the form: Ax ² +Bxy+Cy ² +Dx+Ey+F=0.
 3. A method as defined in claim 1 wherein said polynomial equation is a quadratic equation.
 4. A method as defined in claim 1, further comprising computing a level number and an index of said second tile, creating a new tiling record that includes said computed coefficients, and pushing said new record onto a tile stack.
 5. In a graphics system for processing geometric objects, said graphics system including a spatial hierarchy having a plurality of levels ranging from a finest level to a coarsest level, said spatial hierarchy being organized as a plurality of nested tiles progressing in scale by powers of two, each of said tiles defining a coordinate frame, said geometric objects described by polynomial equations having predetermined coefficients, a method for transforming a polynomial equation from a coordinate frame of one the to a coordinate frame of another tile at a finer level of the hierarchy, said method comprising: a translation computation, and a scaling computation, said scaling computation performed by shifting the binary representation of a coefficient of said polynomial equation or by shifting the binary representation of a polynomial expression including a coefficient of said polynomial equation.
 6. A method as defined in claim 5 wherein said geometric objects are polygons, said polynomial equation is a linear equation, and said polynomial expression is a linear expression.
 7. A method as defined in claim 6 wherein said linear equation describes an edge of a polygon and has the form: Ax+By+C=0.
 8. A method as defined in claim 6 wherein said linear equation describes the depth of a polygon and has the form: z=Ax+By+C.
 9. A method as defined in claim 6 wherein said linear equation describes a color component of a polygon and has the form: c=Ax+By+C.
 10. A method as defined in claim 6 wherein each the includes a plurality of cells and each tile defines a coordinate frame having an origin, said tiles including a first tile at one level of the image hierarchy that corresponds to a cell in a second the in a coarser level of the image hierarchy, the coordinate axes of said coordinate frame of said first tile being x and the coordinate axes of said coordinate frame of said second tile being x′ and y′, the origin of said first tile being (xt,yt) in the coordinate frame of said second tile, said geometric objects including a geometric object which is described in part by a linear equation having the form Ax+By+C=w, where A, B, and C are predetermined coefficients that are relative to the coordinate frame of the second tile, the linear equation in the coordinate frame of the first tile having the form A′x′+B′y′C′=w, said method comprising the steps of: computing A′ by evaluating the expression A/N; computing B′ by evaluating the expression B/N; and computing C′ by evaluating the expression Axt+Byt+C; where N is a power of two.
 11. A method as defined in claim 6 wherein each tile includes a plurality of cells, each tile defining a coordinate frame having an origin, said tiles including a first tile at one level of the image hierarchy that corresponds to a cell in a second tile in a coarser level of the image hierarchy, the coordinate axes of said coordinate frame of said first tile being x and y and the coordinate axes of said coordinate frame of said second tile being x′ and y′, the origin of said first tile being (xt,yt) in the coordinate frame of said second tile, said geometric objects including a geometric object which is described in part by a linear equation having the form Ax+By+C=w, where A, B, and C are predetermined coefficients that are relative to the coordinate frame of the second tile, the linear equation in the coordinate frame of the first tile having the form Ax′+By′+C′=Nw, where N is a power of two, said method comprising the step of: computing C′ by evaluating the expression N(Axt+Byt+).
 12. A method, comprising: computing coefficients of a polynomial equation relative to a coordinate frame of a first tile, utilizing a processor; and transforming said polynomial equation to said coordinate frame of a second tile; wherein an image is displayed based, at least in part, on the computing and the transforming.
 13. A computer program product embodied on a non-transitory computer readable medium, comprising: computer code for computing coefficients of a polynomial equation relative to a coordinate frame of a first tile; and computer code for transforming said polynomial equation to said coordinate frame of a second tile; wherein the computer program product is operable such that an image is displayed based, at least in part, on the computing and the transforming.
 14. A method for transforming a polynomial equation from a coordinate frame of one tile to a coordinate frame of another tile at a finer level of the hierarchy, said method comprising: a translation computation utilizing a processor, and a scaling computation, said scaling computation performed by shifting a binary representation of a coefficient of a polynomial equation or by shifting a binary representation of a polynomial expression including a coefficient of a polynomial equation; wherein an image is displayed based, at least in part, on the scaling computation.
 15. A computer program product embodied on a non-transitory computer readable medium comprising: computer code for a translation computation, and computer code for a scaling computation, said scaling computation performed by shifting a binary representation of a coefficient of a polynomial equation or by shifting a binary representation of a polynomial expression including a coefficient of a polynomial equation; wherein the computer program product is operable such that an image is displayed based, at least in part, on the scaling computation. 